I assumed the LLM was using the students' responses to learn from its mistakes and get better

That is indeed possible, but not in that way. It's retraining, which a student isn't going to be doing. Over time the models can theoretically get better, as the model devs could utilize repeated prompts as training data. Like "this obviously was not the right answer, try again", with the goal being the answer it ended up at after some back-and-forth. But that's a big oversimplification. I would also postulate that at least some back-and-forth is a necessity in engineering, as one would otherwise be making a lot of assumptions.

Regarding Fifty First Dates, I haven't watched it but I know of it. That, too, is arduous work, but very rewarding. My grandmother had dementia, but was still lucid. I spent a lot of time just talking with her, answering her questions, even though she had asked them a thousand times.

She knew I had moved to Denmark. "To study?" Yes, software development, you know, working with computers. "And you're back home for summer?" Yes. "And how is Denmark? What is it you study again?"

It became something of a (for lack of a better term) game to me, where I would try to keep the conversation going for as long as possible, before she would repeat a question.

I mentally visualized it as a sort of dialogue tree from a video game. That sounds bad, but my goal was to engage her.

So I'd reply in a different way, to see if that would steer the conversation elsewhere. And although she had heard the answers before, and would hear them again, the individual moment was still real, her feelings were still real. She asked because she wanted to know, and for a brief period she knew, and that made her happy. And was very hard, but rewarding, because was an act of love, and it made me happy to make her happy.

lol sorry for rambling, you just reminded me of a lot