Alice (not real name) is a student in one of my Math 100 (calculus) classes. It's a course offered by my college as a dual credit course at a high school, so the whole class is about 17/18 years old, and I'm aware that there's sometimes a maturity issue. Still, my high school teacher colleagues agree that Alice is very special.
Alice is really into math -- in her own way. Before or after class she comes to me and shows me something she figured out. She plays around with functions on her graphical program and notices cool things about them when she changes the parameters. She looks up which polynomials approximate the exp function and graphs that and finds it cool. She reads about complex numbers and how trigonometric identities find a natural explanation there. After we covered Newton's method, she'll try to find every zero of every function with it.
And that's where the problem starts: Every zero of every function. In an optimisation problem, where the derivative is an easy parabola, she will not use the quadratic formula to find a critical number, but insists on applying Newton's method. In another optimisation problem she introduces a second variable and wants to find a more general solution, trying to develop multi-variable calculus on her own in the middle of a test, and fails. In a curve sketching question, she does not get beyond the first derivative because she first wants to give a proof of the quotient rule from scratch. In class, she continues to ask about generalisations of the material we cover to the complex setting, even after I have tried to make clear to her that complex numbers are not part of the curriculum and it's important to first get our material straight.
In short, she's very enthusiastic about some mathematics, but only that which catches her attention, and she seems to neglect a lot of the actual material of the course for that. To the extent that she's lost many points on assignments and tests and is at risk to fail the class.
Obviously I've tried to make the issue clear to her, but I see no success so far. I admit it's hard because often she's enthusiastic about things I'm enthusiastic about myself, and I actually like to chat with her about the Riemann sphere and stuff after class. Should I rigorously cut down such conversations?
Also, part of me likes that she thinks outside the box. Where most students' minds are too compartmentalised, or they can solve problems only with a memorised standard method -- Alice tries to use her own approaches, or methods from different sections of the course. Problem is she often makes mistakes then, and it takes so much time that she cannot work on other questions. To a lesser degree, I've had students like that before, and I've always tried to reward original or uncommon approaches, even if they don't entirely work out. But with Alice it's on a new level: She just refuses to use standard approaches even if they are obviously the shortest, most practical etc.; but her own approaches, although never stupid, basically always fail to work out.
What can I do to make Alice pass the course, ideally without crushing her enthusiasm for mathematics?
Edit: Maybe I should clarify the following: While I do think that Alice has talents that are underappreciated, as far as I can tell she is not a hidden genius or savant. When I say "we chat about the Riemann sphere", it's not like she has a deep understanding of complex geometry, rather like she read about it on a Wikipedia level, understands the basic projection idea, is all fascinated about having a way of putting "infinity" into the numbers. (Not in a crackpot "I can divide by zero" way, but not in a profoundly insightful way either.) I can tell she misunderstands certain concepts, then I try to talk her out of it, which sometimes works and sometimes doesn't. Her approaches are always original, but also often flawed, sometimes for reasons that are obvious to anyone with a formal math education. She still makes basic mistakes. The point I admire about her is the genuine fascination with, and enthusiasm for, mathematics, as well as her potential for original thinking; but her thoughts show a lack of precision and rigour.
My question is not how to save a new Ramanujan from a hostile conformist education system. It's how to help a student getting their priorities straight without crushing their curiosity and motivation for self-learning.