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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.

Torsten Schoeneberg
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    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] – paul garrett May 03 '18 at 23:29
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    ... [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... – paul garrett May 03 '18 at 23:30
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    Also, it occurs to me to be clearer in a certain way: "please send her here to our department!" :) I think you see what I mean. – paul garrett May 03 '18 at 23:31
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    I personally don't think her enthusiasm is "wrong"; some might think that it is even "right." Perhaps you should consider using the word "different," "unusual," or "nonstandard" instead. – JRN May 03 '18 at 23:40
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    Also, it is very much worth noting that there is "mathematics", and then there is "mathematics as a school subject". Things we do for pleasure versus things we do to run gauntlets. I tell my grad students to please try to stop thinking of mathematics as a school subject at all, but, instead, as a thing real people do in the real world, with no exams, etc. – paul garrett May 03 '18 at 23:46
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    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. – paul garrett May 03 '18 at 23:49
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    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. – Adam May 04 '18 at 00:55
  • @JoelReyesNoche: Her attitude is wrong in the precise sense that it is likely to get her an F on her academic transcript. As I wrote, I'm aware part of my problem is that even I would consider her attitude "right" in a different context, that's why I put the "wrong" in quotation marks. – Torsten Schoeneberg May 04 '18 at 02:19
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    @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.) – Torsten Schoeneberg May 04 '18 at 02:25
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    Is she brilliant but willful? Or is it possible that she knows how to do Newton's method and has a hard time learning the other stuff (weaker intrinsic ability)? The first might be some sort of strange savant (still a problem, to deal with). The second would be someone who is hiding from dealing with their weakness by doing what she knows. (also a problem to deal with, but perhaps a different solution). – guest May 04 '18 at 02:43
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    As a possible suggestion, if there is any way you can get her to help a weak student after class or grade homework, she would be forced to focus on one topic for an extended period. Of course, helping a weak student probably isn't good if she is too much of a "wandering spirt" (gets off on tangents, lacks focus, etc.), and grading work is probably not something she should do, but I mention these as activities that you might try to simulate by the use of other activities. For instance, maybe you could have her proof-read and double check some worked out examples you plan to present to class. – Dave L Renfro May 04 '18 at 08:52
  • @DaveLRenfro: Interesting suggestions. I'm afraid she is indeed too much of a wandering spirit for the first approach, and it quite probably would do more harm than good. But this "could you check these examples for me, please, I want to show them next week" might motivate her and guide her to do what she's supposed to do, I like it a lot. – Torsten Schoeneberg May 04 '18 at 18:50
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    @DaveLRenfro Make it an answer! – Chris Cunningham May 04 '18 at 19:01
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    @Chris Cunningham: It might be tomorrow before I get to this, as I'm really busy with something this afternoon. However, by then maybe I can also think of other ways to "simulate" those activities. – Dave L Renfro May 04 '18 at 20:08
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    @Chris Cunningham: A project I'm involved with this week (day job stuff) will keep me from dealing with this for another few days. I did make time for something else yesterday (couldn't help myself!), but I probably shouldn't have . . . (I have been jotting down some ideas as they come to me, however.) – Dave L Renfro May 09 '18 at 09:03
  • I think your response could be colored by her attitude toward failing the class. I feel she is young enough that retaking a college class when she is older is not the end of the world. If she doesn't care about her grade, then give her as much freedom as she wants. Perhaps in time she will learn to focus; the issue is avoiding any stifling. An alternative would be to see if she could get an independent study class that would allow her to pursue tangents as much as she wants; the compromise she would need to make to take it would be to focus on the material she needs to learn in your class. – scott May 09 '18 at 19:55
  • @paulgarrett: I appreciate your comments, and agree with most of them on a math-lover level, but I admit I don't find they address my main question: How to help her pass? I'll happily give her your email after the final exam, but as of now, I am convinced that her starting a conversation with you (about ...?) would be counterproductive. Please understand, this is not meant as doubt about you, it's just my assessment of the immediate situation and what's helpful for her right now. If you disagree, please expand your suggestions in an answer, or we move the discussion to chat or private email. – Torsten Schoeneberg May 11 '18 at 22:50
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    @TorstenSchoeneberg, ah, of course, use your own best judgement. You are there, and I am not. I'd just recomment (though kids often cannot understand this distinction, not to mention adults) that short-term conformity may be useful or essential, but is NOT the same thing as long-term conformity. – paul garrett May 11 '18 at 23:24
  • Did she pass??? – Joe Oct 05 '19 at 22:36
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    @Joe: Barely, yes. – Torsten Schoeneberg Oct 05 '19 at 23:18
  • "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. " Funny. With the current popularity of machine learning and the ubiquity of "gradient descent" in machine learning, many professionals in applied mathematics make the same mistake as Alice and use gradient descent to find zeroes which they could have calculated directly from a simple equation. – Stef Aug 12 '22 at 07:20
  • @Stef: It's not funny. This kind of behaviour is truly a hallmark of incompetency. Often it exists merely because society tolerates such incompetency, partly because the ones in charge are not that competent themselves. Sometimes it is at the more extreme end of the spectrum, in which case you get one type of crank. There are in fact many cranks on SE that arise due to this, pretending to know things that they copy-pasted from wikipedia, or pretending to know mathematics intuitively but without being able to actually do real mathematics. – user21820 Jun 01 '23 at 12:37
  • Encourage promising genius. – End Anti-Semitic Hate Jun 20 '23 at 08:41

5 Answers5

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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.

Chris Cunningham
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  • Great way of putting this in the boldface. I'm likely to link back to this a lot in the future. – Daniel R. Collins May 04 '18 at 15:56
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    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. – Chris Cunningham May 04 '18 at 17:40
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    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." – Benjamin Dickman May 05 '18 at 04:57
  • 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. :-) – Daniel R. Collins May 05 '18 at 08:10
  • +1. There is no shame in counting on your fingers. Better a humble grounding in truth than a torrent of glittering sophistry. – nanoman Jun 07 '21 at 06:50
  • From a very early age, we accept, for example, that 1+1=2 without any formal proof. That is not to say that a formal proof does not exist. Here is my formal 791-line proof: https://www.dcproof.com/ConstructAddFunctionV2.htm It would be insane to require such a proof at the high-school level, or even in a university-level pure-math program. There is certainly something to be said for memorizing a few basic results without requiring a detailed understanding of their proofs. Put another way, carpenters need not know to construct a hammer and saw to be competent in their trade. – Dan Christensen Feb 24 '24 at 03:53
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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).

Benjamin Dickman
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  • She might like other books about problem solving too. Paul Zeitz's The Art and Craft of Problem Solving might be good. – Sue VanHattum May 09 '18 at 00:56
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    @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. – Benjamin Dickman May 09 '18 at 01:08
  • 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. – 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?)

Dominique
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On a practical level, give extra credit assignments about things she is enthusiastic about. Give generous partial credit on these assignments for original ideas that don't necessarily work out. Let her pass the course based on the marks you give her for those assignments.

Kostya_I
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  • Why would you do this? This is not at all how I would approach the situation (this is fine of course), but you haven't included any reasons for taking your approach so it's difficult to be convinced by this. – Chris Cunningham Feb 23 '24 at 18:35
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    @ChrisCunningham, the OP did not ask for opinions about how to approach the situation; there was a specific question that read: "What can I do to make Alice pass the course, ideally without crushing her enthusiasm for mathematics?" My answer gives the simplest way to do just that. – Kostya_I Feb 23 '24 at 21:52
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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).]

guest
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    I cannot condone this attitude. It only advocates conformity, and is nothing about real mathematics. What is the goal of "school"? – paul garrett May 04 '18 at 00:31
  • If you want to help her learn the subject (this is her immediate task and her teacher's job), you will do something similar to what I say. If you are OK with her failing, then spend more time on Newton's method and the Reiman sphere and keep blowing off what she is supposed to learn. And if you think that's harsh, it's not. It's life. Work world is the same. There are amazing things you can do and lots of fun to be had...but not if you don't get your assigned work done first. – guest May 04 '18 at 00:47
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    (-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? – Brendan W. Sullivan May 04 '18 at 00:49
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    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. – guest May 04 '18 at 00:52
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    For your last comment, yes. However, "the beating will continue until morale improves" is not really an effective strategy. – Adam May 04 '18 at 01:21
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    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. – guest May 04 '18 at 15:05
  • I've actually upvoted this answer (as all the others), although I agree with @Adam's comment, and it does not address the "without crushing her enthusiasm" part of my question (or rather says, I should not aim for that). And, "slamming her on the tests" is not going to bring her closer to passing because the test marks count towards the final mark. So this is not the approach I would like to take, but I also think that people should only downvote it if they have a better suggestion. – Torsten Schoeneberg May 04 '18 at 18:56
  • Thanks Chris. Even though my solution is unpopular, I actually probably have directly answered the request. Rather than general commentary. Usually I am the opposite. P.s. I have had 3 upvotes and 8 downvotes. I'm at -5 for the indicator but +14 for gold sticker bonus point rating: 310 - 82 = 14. 1064 for overall rating...pretty good for a bad boy of the classroom! ;) – guest May 04 '18 at 18:57
  • @guest: It is an answer to the actual question (how to A. make her pass without B. crushing her enthusiasm), but it neglects part B on a meta-level, and I'm not convinced it would help a lot for part A either. – Torsten Schoeneberg May 04 '18 at 19:02
  • TS: Thanks, big guy! Yes, I haven't answered the "how to do A without doing B" but I think parts of your question included the option for not worrying about the B any more or that it was actually interfering with the A: "Should I rigorously cut down such conversations?" [Uh...duh!] Anyway, good luck. And remember this is not a topology problem, but a leadership and coaching and training problem. (Oh...and this is MUCH more in the sweet spot of the forum, shouldn't have worried about that...than many questions which are math questions hiding in pedagogy. so go you!) – guest May 04 '18 at 19:02
  • TS: It would be really strange if you accepted my answer. I could have the grey flashing hard to read spanking AND the green helmet pat on the head sticker. That would really crash the cars! ;-) Isn't the Internet fun? (Don't ban me, quad.) – guest May 04 '18 at 19:08