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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.

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    $\begingroup$ Apart from trying to answer your immediate question... it is wonderful that your student has such enthusiasm! In my own trajectory, although I was fairly precocious in mathematics and had always excelled in all classes, including math, at 10th grade in high school in the U.S. for some reason I balked at conforming to the expectations of the math classes (mostly memorization and specified use of symbols) even though I'd learned calculus and other things. Almost failed "Algebra Two" for that reason. But... [cont'd] $\endgroup$ – paul garrett May 3 '18 at 23:29
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    $\begingroup$ ... [cont'n] exactly because I was intensely engaged with mathematics (on my own terms, yes), things turned out ok in the long run. Don't accidentally stifle your student! Such fervor is rare! We know what her destiny is, I think... $\endgroup$ – paul garrett May 3 '18 at 23:30
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    $\begingroup$ For that matter, please give her my email address, and encourage her to send questions or assertions, etc. I do want to encourage young people who may not be fitting into the usual conformity-intensive academic math context (esp., in the U.S.) Her attitude is not only compatible with "real math", but, also, while dooming her to be a short-term academic mis-fit, augurs well for her future as a mathematician. $\endgroup$ – paul garrett May 3 '18 at 23:49
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    $\begingroup$ Is there any indication that the student is not "neurotypical"? Such students may be eligible for special accommodations so that they can get through the standard curriculum -- which she needs to do so that she can more easily continue with that part she's most interested in. $\endgroup$ – Adam May 4 '18 at 0:55
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    $\begingroup$ @Adam: Definitely. She has access to an IEP from her high school, which practically means she could get more time for writing tests. This is certainly something I would be willing to try. However, so far she has refused (!) to take that option. Also, my speculation is she would not use that extra time well either. (On the latest test, when I asked to apply Newton's method graphically two times, she spent ten minutes or so drawing more tangent lines until the graph was filled up, all the while not working on other questions.) $\endgroup$ – Torsten Schoeneberg May 4 '18 at 2:25
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My answer is maybe a little bit off. Still, I have had some luck in the past with two separate similar students to yours by communicating roughly the following concept using the outline below.

An expert knows both the theorem and the proof.

Ask this student what is $3 \cdot 5$. Then ask the student how they know for sure that this is the right value of $3 \cdot 5$. An expert multiplier will both know the answer is 15 and that the reason this works is because you can make three rows of five dots and count them up. Probably your student is an expert multiplier (If it turns out she is not, move back to addition and use $3 + 4$).

Establish with the student that it would be possible for a student to make it through elementary school in either of the following two unfortunate situations:

  • The student has memorized that $3 \cdot 5 = 15$, but does not know why and would not be able to convince anyone who had memorized it incorrectly.
  • The student has not memorized that $3 \cdot 5 = 15$, and when presented with this problem, always draws all the little dots and counts them up.

Both of these situations would be bad. Try to get her to explain why they would be bad.

Explain to the student that neither one of these hypothetical situations spells permanent doom. A student who refuses to learn reasons can eventually be convinced to shift focus toward reasons. A student who refuses to buckle down and learn facts can eventually be convinced to shift focus toward facts.

Usually we end up needing this idea to tell students that they should seek understanding beyond mere shortcuts and cookbook methods that they are interested in learning. Tell the student this. It will be useful to engage the student where she is, which is that she thinks learning facts and efficient methods is boring. She knows she is exceptional and it is useful and good to acknowledge this. You could even go so far as to say that of the two unfortunate situations we talked about, the second one is better because it has a higher likelihood to lead to success in the end.

Anyway, at the end you need to tell her that she is in danger. That she is too focused on one part of math -- the part of math that most students find uninteresting, which is really good and awesome -- but that she could fall through the cracks and needs to refocus.

That in the end, it's cool that she is interested in the proofs, but can't spend her whole life multiplying numbers using arrays of dots. She has to learn the theorems too.

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  • $\begingroup$ Great way of putting this in the boldface. I'm likely to link back to this a lot in the future. $\endgroup$ – Daniel R. Collins May 4 '18 at 15:56
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    $\begingroup$ I'm pretty sure I learned this phrasing on this site (it's not mine originally), but I couldn't find the reference to it. If someone does, I'll link it. $\endgroup$ – Chris Cunningham May 4 '18 at 17:40
  • $\begingroup$ The closest I could find on MESE is from paul garrett here: "Authority is not the same as expert-ness. An expert can persuade by showing how wonderfully a thing can be done." $\endgroup$ – Benjamin Dickman May 5 '18 at 4:57
  • $\begingroup$ I may be biased because it sounds like something I'd say to begin with. There's at least a very small chance I said something a lot like that in the past here. :-) $\endgroup$ – Daniel R. Collins May 5 '18 at 8:10
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First, a response to a particular excerpt:

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?

Stepping back for a moment: This is a situation in which a student is enthusiastic about mathematics, you are enthusiastic about mathematics, and you are asking whether to stop conversing enthusiastically with her about mathematics. To this I say: No! How wonderful it is that you have an opportunity to engage meaningfully with a student whose interests in mathematics are both burgeoning and variegated. Keep the conversations going.


Back to the remaining issues: Alan Schoenfeld has a nice book called "Mathematical Problem Solving" (1985) and another called "How We Think" (2010) that I've mentioned elsewhere on MESE in the past. The gist of his ideas, for the purpose of this question, boil down as follows: problem solving (in mathematics) involves beliefs (e.g., how long does a typical problem take to solve?), strategies or heuristics (e.g., quadratic formula, Newton's method), and metacognition (thinking about thinking in the context of problem solving - e.g., how should I deploy my strategic resources in order to make headway on this problem? or: I've been trying a particular strategy for a while; should I adjust?).

The student that you are describing seems to be doing well with beliefs (e.g., she is enthusiastic about mathematics, wants to discuss mathematics, believes herself capable of engaging with mathematics) and seems to be interested in various strategies (even delving into material that is not formally part of your course). But, I think that the metacognitive component is where some issues are arising: An important part of mathematical thinking is matching problems with problem solving methods, and (in my interpretation) that means picking an approach as a function of the problem. (So, not picking the approach - e.g., Newton's method - beforehand, and then trying to apply it willy-nilly irrespective of the problem at hand.)

Here is my concrete suggestion (and maybe I will reedit this later if you update your question, or leave a comment, etc): Since the student seems not short on energy/enthusiasm, see if you can get her to read the article "What's All the Fuss About Metacognition?" mentioned in my MESE answer here. If you cannot find a full copy, then let me know and I'll upload one. Then, see if you can engage with the student about mathematics not being just strategies/resources (which she seems to be accruing impressively!) but also thinking about how they are used when faced with a particular mathematical question for which the method of solution is unknown at the outset (i.e., attending to metacognition: fitting a method to a problem rather than the other way around).

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  • $\begingroup$ She might like other books about problem solving too. Paul Zeitz's The Art and Craft of Problem Solving might be good. $\endgroup$ – Sue VanHattum May 9 '18 at 0:56
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    $\begingroup$ @SueVanHattum Maybe (and you could, of course post an answer to this effect!) but my recommendation is not to find additional problems to solve, but rather to understand the research around problem solving and the important role played by metacognition. I am not, in the answer above, suggesting mathematical readings (texts, articles, etc) that go further yet from the curriculum at hand. $\endgroup$ – Benjamin Dickman May 9 '18 at 1:08
  • $\begingroup$ No time for a whole answer. Your recommendation sparked mine because Zeitz is metacognitive too, and because she might do better in class if she thinks about things from the perspective of problem solving. None of us know her, and I trust that she will eventually find her way. Something will help her shift. $\endgroup$ – Sue VanHattum May 10 '18 at 15:02
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I envy you: having such a student is very, very interesting and pleasant, but obviously, if she fails due to this immature behaviour, she (and you) will have a problem.

The first thing which comes in my mind: how smart is she?
Let me explain: you say that, for solving a simple parabola, she uses Newton instead of simple quadratic solving formula.
So why not use both? She can use the Newton way, use the quadratic formula, and be amazed that both are converging to the same result.

This approach (using her enthusiasm for leading her to the actual course content) is only applicable if she is smart enough to cover both in a reasonable timeframe. Is this the case? (and can you even know that? Are you sure she's not neglecting her other courses because of her interest in mathematics?)

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"She seems to neglect a lot of the actual material of the course...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?"

  1. Stop encouraging her behavior. If she were getting done what she needs to get done, then fine, no problem. But her interest in estimation is interfering with mastering basic learning objectives for analytical solution.

  2. Discourage the behavior.

  3. Get the parents involved.

  4. Slam her on the tests.

[If it still doesn't work, oh well. You can't save them all. And it is her life. People fail classes for a lot of reasons (ability, laziness, disinterest, and misprioritization).]

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    $\begingroup$ I cannot condone this attitude. It only advocates conformity, and is nothing about real mathematics. What is the goal of "school"? $\endgroup$ – paul garrett May 4 '18 at 0:31
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    $\begingroup$ (-1) The fact that you outlined "stop encouraging her behavior" and "discourage the behavior" as two separate steps shows that you really want to quash her desire to understand how she is using mathematics. Why not devote all the time/effort it would take to "discourage" her behavior towards pushing that behavior in a different (and perhaps, as you see it, "more productive") direction? $\endgroup$ – Brendan W. Sullivan May 4 '18 at 0:49
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    $\begingroup$ She is spending time on the wrong stuff and it is interfering with getting main objectives accomplished. I have no problem with the enrichment if it is on top of getting basics done. But it is actively interfering with it. So we should change that. $\endgroup$ – guest May 4 '18 at 0:52
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    $\begingroup$ For your last comment, yes. However, "the beating will continue until morale improves" is not really an effective strategy. $\endgroup$ – Adam May 4 '18 at 1:21
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    $\begingroup$ I think a mixed message is being sent with all these attention to her interests that are not on topic. Active harm is being done with this mixed message. That should be cut off completely until she gets up to speed on the basics. (If she still persists on her own, that is one thing, but don't encourage it or foster it in any way.) Also the "beatings" haven't really started yet. Maybe if this stuff was nipped in the bud earlier, we wouldn't have this situation. $\endgroup$ – guest May 4 '18 at 15:05

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