The Surprising Simple Math Behind Puzzling Matchups — Quanta Magazine

My latest column for Quanta Magazine is about one of my all-time favorite mathematical ideas: transitivity. Well, technically it’s about intransitivity, a subtly complex mathematical situation which any sports fan knows all about.

It’s the championship game of the Imaginary Math League, where the Atlanta Algebras will face the Carolina Cross Products. The two teams haven’t played each other this season, but earlier in the year Atlanta defeated the Brooklyn Bisectors by a score of 10 to 5, and Brooklyn defeated Carolina by a score of 7 to 3. Does that give us any insight into who will take the title?

Well, here’s one line of thought. If Atlanta beat Brooklyn, then Atlanta is better than Brooklyn, and if Brooklyn beat Carolina, then Brooklyn is better than Carolina. So, if Atlanta is better than Brooklyn and Brooklyn is better than Carolina, then Atlanta should be better than Carolina and win the championship.

Sports fan knows things are never this simple, and in my column I explore some of the surprising mathematical reasons why it may be the case that A is better than B and B is better than C, but C is better than A. You can read the full column for free here.

My Tests are So Hard

Everywhere I’ve taught there have been teachers who brag about how hard their tests are. It’s always a central part of their identity as a teacher, of how they see themselves, and how they want to be seen. They proudly consider themselves more rigorous than their colleagues.

But nothing could be easier than making a test hard. You can just put more questions on it than can be reasonably handled in the allotted time. Or put problems on that haven’t been emphasized in class or practiced enough. Or problems that test edge cases and not core ideas. Or problems from the next unit. Or problems you simply haven’t prepared all students to handle.

I’ve seen teachers do all these things. It’s not rigorous. It’s lazy. You know what’s truly difficult? Writing a test that is fair, representative of core ideas, and appropriately challenging.

Taught Helplessness

I’m currently reading “The Design of Everyday Things” by Don Norman and it’s interesting to think about what the theory of product design has to say about instructional design.

For example, the author discusses how “learned helplessness” can result from poor design. A product whose functionality isn’t discoverable, and that doesn’t provide good feedback, will be frustrating to use, so users will likely give up after trying and failing a few times.

Just as I was making the connection to teaching math in my mind, the author himself brought up math instruction as a common example of “taught helplessness”: When math is presented as unintuitive, and poor or misguided feedback is given, students are likely to just give up. The problem is amplified by the linear way in math is usually taught. In many classrooms, if you don’t understand what happened yesterday, you will probably struggle to understand what is happening today.

Originally posted on Mastodon.

2023 — Year in Review

As I reflect on my professional 2023 I keep coming back to the line “Normal is normal again.” Not the world, certainly, and not for everyone, but since changing schools nearly five years ago this is the first year when being in the classroom hasn’t felt extraordinary for extraordinary reasons.

Still, it’s been a mix of the new and the old. In addition to my normal lineup of geometry, calculus, and linear algebra courses, I’ve been doing more mathematical modeling with students this year. In the spring we had a team of modelers invited to compete in the International Mathematical Modeling Contest (IM2C), thanks to their excellent work on last year’s High School Mathematical Contest in Modeling (HiMCM), and I’ve been working to expand the program at school. This year interest was high enough to have eight teams compete in the HiMCM, and we’re looking forward to new modeling opportunities this spring. In addition, a colleague and I mentored around 70 students who competed in a national forecasting tournament inspired by the Good Judgement project, and our teams took first and second place!

An extracurricular highlight this year was interviewing mathematician and author Manil Suri for the Jaipur Literature Festival. We had a lively talk about his excellent new book, The Big Bang of Numbers, and also about math, writing, and teaching, and the Asia Society of New York made the entire video of our conversation available here.

As usual, I continued to design and run workshops for teachers this year, including a new entry in my ongoing linear algebra series titled Learning to Love Row Reduction. I also gave two talks at this year’s NCTM Annual Meeting: So, You’re Teaching Pre-Calculus, and A Case for Linear Algebra. I’m already looking ahead to new talks this coming year, including an upcoming workshop on the geometry of linear regression I’ll be presenting in February.

Writing my column for Quanta Magazine was as challenging, and fulfilling, as ever, with pieces aimed at bridging the gap between classroom and research math focused on the newly discovered aperiodic monotile, the algebra of secret codes, graph theory and cliques, a high school student’s amazing proof, and what three-point turns tell us about a hundred-year-old geometry problem.

And with the landscape of social media continuing to change, I’ve been enjoying my time on Mastodon more and more in 2023.

Platforms based on decentralization, user autonomy, and interoperability definitely seem like the right way forward. And I’ve been trying to do a better job of archiving what I write by cross-posting some of my social media posts here on my blog.

It’s been a good 2023, and here’s hoping for another good, and relatively normal, 2024!

Related Posts

ChatGPT in Geometry Class

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.

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