Friend of mine just asked me for an old twitter thread I wrote one time about how I turned a “cheating” moment into a learning moment. I had to dig it out of the json files in my twitter archive (relatedly, if anybody knows a good way to pull a thread out of the twitter archive, rather than just individual tweets or replies, hit me the fuck up), so I thought I would post it here on The Blog for accessibility slash future reference.

The thread!

I'm sitting at the local Jimothy Johnathan's waiting for my sandwich, so: HOW SOMEONE WHO IS FUNDAMENTALLY OPPOSED TO COP SHIT HANDLES “"”CHEATING”””, an thread with inevitable interruptions (1/n)

I assigned students in my intro real analysis class to prove the triangle inequality. This is hard! I wasn't really expecting complete proofs out of anybody the first time around and tried to communicate that to them: “just give me a first shot,” I said. (2/n)

One student submitted a beautifully-written and entirely correct proof. Unfortunately, it was for the version of the triangle inequality that involves moduli of complex numbers. (3/n)

SANDWICH BREAK (4/n)

a cartoon sandwich with a big smile dancing happily on a plate

Anyway, so you can totally clock the issue from a mile away, because the proof for complex numbers involves multiplying (a+b) by its conjugate, which is represented with a bar over the quantity. (5/n)

As soon as I saw the proof I knew it wasn't the student's work, because I know for a fact that this student has never taken complex analysis. (We haven't offered complex in five years; I was going to offer it the semester it got canceled; I'm still salty.) (6/n)

So when I was meeting with the student today (for an unrelated reason), I asked them, “btw, can you talk me through your proof of the triangle inequality?” Obviously I already know that the answer to this question is, no, they can't. (7/n)

The student said some various weird stuff about “you know, when you're doing a quadratic, and you multiply (a+b) by itself…” and I replied, “So what does this bar here mean, then?” Obviously I know that the student does not know the answer to this question. (8/n).

This is a question that is intended to provoke dangling. This sounds like I am on a power trip, but I am not. It is useful for the student to face, inescapably, the fact that they do not understand the work they submitted. (9/n)

The whole reason why I, a person who fundamentally does not believe in cop shit, still want to check a student about “"”cheating””” is that it is work that does not produce learning. My whole thing is learning, and “"”cheating””” short-circuits this process. (10/n)

Anyway, the student did indeed dangle. “It's like when you have a matrix.” A matrix, eh? “Yeah, like… uh…” dangling silence. (11/n)

Satisfied that the student has recognized that they do not understand the work they submitted, I now let them off the hook. “So let me put my cards on the table, I know what this bar means, and it's a thing called the complex conjugate.” (12/n)

Do you know much about complex numbers, student? “Not really.” Okay, so they're like z = u+vi, and we can draw them on the complex plane, and here's the real axis and the imaginary axis, and we can identify z with the point with coordinates (u,v). (13/n)

The complex conjugate z-bar is replacing the imaginary coordinate with its negative, and that corresponds to flipping the point over the real axis.
As I'm giving this explanation, I can tell that the student is interested and tracking. This is good! It's what I wanted! (14/n)

I can tell because at this point the student goes, “So it's like the opposite.” Yes! Like, the imaginary opposite. Okay, so now check out this cool thing that happens when you multiply z by zbar: the cross-terms cancel, the i squares out to -1, and you get u^2 + v^2. (15/n)

Here's why that quantity is cool, is because if you think of the complex number as a vector, you can see that's the square of the length of the vector, which we call the modulus. And now, annoyingly, we use the same |•| notation for modulus that we use for absolute value. (16/n)

The student now realizes the issue. (17/n)

Which is crucial, right? It's one thing to say to a student, “this can't possibly be your work because it involves things that you don't know what they are,” and it's another thing entirely for the student to be like, “oh, I didn't know what that was.” (18/n)

So now when I say, “this can't have been your work,” an interesting thing happens:
THE STUDENT AGREES: “Yeah, I got help.”
By taking the opportunity to actually teach the student the cool thing they used without understanding, I've given them cover to agree! (19/n)

So then there's the usual denouement, right: “this is work you didn't learn from, it defeated the purpose of doing the homework, please don't.”
But instead of a lecture, it's an articulation of a mutual understanding. This is a non-negative experience for the student! (20/n)

The student said, “Yeah, I got help, but it wasn't helpful. It was the wrong help.” Yes exactly! And because that's something they said, directly out of their own understanding of the situation, it's way more meaningful to the student than if I said it at them. (21/n)

AND, finally, because I haven't sabotaged my relationship with the student by being a cop at them, I get to tell them, “please come to me for help next time because it's literally my job to help you understand this material,” and they get to believe me. (22/n)

So what have we learned today? Even if you don't believe in cop shit, you can still address cheating. You can let a student dangle, because it serves an important purpose. Look for opportunities to teach cool things about complex numbers or whatever. (23/n)

And finally, and most importantly, whenever possible, build relationships rather than destroying them. Then you get to keep teaching, instead of being forced to keep being a cop.
That's all. I love you. (24/24)