AI-Generated Letters of Recommendation

I write around 30 recommendation letters a year. These are mostly for students applying to college, but increasingly I’m asked to write recs for competitive summer programs, private schools, scholarships, even internships. It’s a lot of work. I estimate that I spend around 100 hours a year on it

And it’s uncompensated work. Almost all of these hours come directly from my personal time, which colleges treat as a free resource. There is nothing to stop them from making me fill out one more form, complete one more ranking, respond to one more school-specific prompt. I often feel like collateral damage in the school admissions arms race.

Some teachers simply refuse to do it. I have come to empathize with that position, but ultimately these recommendations are important to my students, so I put in the time and effort, even though the process is frustrating.

What’s most frustrating is that I’m not sure all this effort makes any difference. Do letters of recommendation really matter in college applications? I find it hard to believe they do. Last year 30,000 students applied to MIT. Who reads those 60,000 letters of recommendation?

I’ve long assumed that these letters just get passed through some kind of sentiment analysis software, where a large language model produces a score, appends it to the student’s profile, and the admissions process grinds on, one automated step at a time. I even recently speculated that colleges were feeding my letters to LLMs without my consent. What’s to stop them?

So when I logged into Naviance, the now-universal portal for college admissions, I shouldn’t have been surprised to see a new feature under the “Letters of Recommendation” tab: a compose-with-AI button.

But I was surprised. Isn’t this an admission that letters of recommendation aren’t that important? If colleges will accept an algorithmically-generated, averaged-out narrative as a substitute for whatever I might have said, how could they possibly value what have I say? Why shouldn’t I just click “Compose”, fill in a couple of blanks, and reclaim my time?

I guess there’s a part of me that still believes a good letter of recommendation can have an impact. Maybe that’s naïve, but if it’s true, then anything less than my full effort would put my students at a disadvantage. I respect them too much to do that, even if the process doesn’t respect me.

For now, I’ll hope that my carefully considered letters will give my students an edge in a world of AI-powered chatbots processing AI-generated recommendations. But I’ll be watching this AI-powered arms race closely.

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Good AI, Bad AI

At the start of the school year I asked my students to let me know how they are using AI in my courses. I’ve seen some Good AI, and some Bad AI.

Good AI

I make mathematical computing a part of my year-long Linear Algebra course. The varying levels of computer programming experience among my students makes this a challenge: some could intern at Google and some can’t remember how a for loop works. AI Coding Assistants, used appropriately, provide invaluable support for students with limited experience. If they can’t successfully write a program to add two rows of a matrix together, they can have the Coding Assistant do it, check to see if it works, and then review the code themselves and try to learn something. Good AI! Here, “appropriate use” means making an honest effort to complete the challenge yourself first: This builds context for learning from whatever code the AI produces, and it also better positions the student to evaluate whether or not the code actually does what they asked it to do.

Bad AI

At the beginning of my Calculus course I ask students to write about a “mathematical observation” they’ve had. I am intentionally vague about what a “mathematical observation” is. Some students write about analyzing their commute to school, some about optimizing a video game strategy, some about a number theory course they took in a summer program. One goal of the assignment is to learn about my students as individuals and as mathematicians, so knowing what they think constitutes a “mathematical observation” tells me something about them.

One student began their paper by disclosing that they first asked ChatGPT to define “Mathematical Observation” for them. This immediately struck me as Bad AI. Thinking about what constitutes a mathematical observation was the point. Not only did the student ask an AI tool to do their thinking for them, but doing so undermined the very purpose of the assignment: Instead of learning what the student thinks as an individual, I got some averaged-out sentiment from a non-random group of authors.

For the record, the student did write a lovely and thoughtful mathematical observation, but afterwards we had a good conversation about the role of AI tools. “Don’t use AI to do your thinking for you,” I said, which seems like a good place to start in navigating this new landscape.

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.

Teaching Triangle Angle Sum

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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Beyond My Control

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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