I'm helping out my former calculus teacher as a volunteer calculus advisor, and I have under my supervision 5 students. They've already had an exam and... well, they failed.
I read their exams and I notice that they did understood the concepts, but when it comes to writting mathematically they lose it, even if they know exactly what to do informally, when they are trying to write it down it seems like they are trying to speak russian without knowing the language yet insisting to speak. Just to give you an example of what I mean, one of them said that given a $w\in\Bbb R$ because something happend then $w$ was countable, even tought a lot of times in class I couldn't be more emphatic that being countable was a quality for sets, and they reapeting it each time like so.

I'm not trying to mock them, not at all, I'm really concerned about them, and since the class has started to get very mathematic, I'm afraid that they might get depressed or something.
What's next class-wise, is to get used to the concepts of supremum and minimum. They already have the definitions, and they can reapeat them perfectly, however I don't think they know what they mean, I want to show them with examples a little bit more flashy but I'm affraid that they might not understand, so what do you recommend?

marginal note: I'm an advisor, means that I'm not the teacher, I'm like a helper, but in the words of my teacher "more reacheble and with less responsabilities, since I'm also a student". Also when we have see each other, they supposedly already have the great class of the professor.

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    $\begingroup$ In your custody...do we need to send you money before you release them? Kidnapping is a serious offense. $\endgroup$ – Will Jagy Oct 3 '13 at 0:20
  • $\begingroup$ @WillJagy oh god... I didn't saw that coming. I just changed that word. $\endgroup$ – Ana Galois Oct 3 '13 at 11:50

You're an adviser so advise them well. You say that they have trouble writing mathematical statements, so you can help them with things such as these. If you want to show them that countability is a property of sets, show them this. Justify the claims that you make to them by at least giving examples, small proofs, or arguments. If they do not understand your reasoning, it could be that they haven't had exposure to formal logic. In that case, show them a bit of formal propositional logic, (If it is worth the trouble). It is obvious that you care about their experience so, at the very least, give them resources that justify some of the things that they need to understand. It could be the problem that they are anxious around others and need a good book to read and privacy to ingest some of these concepts. At the end of the day, if they aren't willing to buckle their seat belts and agree to do whatever it takes, they are not interested in understanding but rather, they have other motives. In this case, it is definitely beyond your obligation to inspire them. If it behooves you to motivate them in some fashion, they need to know that you care about them AND you need to have the ability to make the material engaging and interesting.

  • $\begingroup$ (1) The fact that countability is a property of sets is not something to be proved: it’s inherent in the definition of the property. (2) Lack of exposure to formal logic is almost certainly not the problem, and exposing them to formal logic is more likely to increase their confusion than to diminish it. $\endgroup$ – Brian M. Scott Oct 3 '13 at 6:39
  • $\begingroup$ You just don't like the way I expressed my ideas. Certainly exposure to formal propositional logic cannot detract from anything. They are already confused, I'm merely suggesting that wherever the confusion lies, there's a strategy for making things more clear. I make no argument for whether or not it is a good strategy. If you have something that might supplement my answer, I would appreciate it, (beyond describing how it is inexact or simply disagreeing with some of my claims). $\endgroup$ – Rustyn Oct 3 '13 at 7:04
  • $\begingroup$ No, I’m not just objecting to the way in which you expressed your ideas: I think that the two that I addressed are simply wrong. The first one certainly is: the fact that countability is a property of sets is not something that can be proved. And I know from a great deal of experience that exposure to propositional logic at this stage can hurt and often does. $\endgroup$ – Brian M. Scott Oct 3 '13 at 7:09
  • $\begingroup$ I didn't mean to imply that the countability is a property of sets was something to be proved in the first place, it was a mistake in the way that I wrote up my response. What I really meant was that one could exhibit bijections between sets and the naturals numbers and in some sense (prove, or show) or whatever have you that that set has the property of being denumerable. $\endgroup$ – Rustyn Oct 3 '13 at 7:14
  • $\begingroup$ The idea is that if they are already being exposed to statements that have logical operators, they should be able to discern the truth of these statements given the truth value of their connectives. $\endgroup$ – Rustyn Oct 3 '13 at 7:21

If these students really do understand the concepts and are not simply parroting the definitions back to you, the underlying problem is most likely that they are completely unaccustomed to expressing themselves with precision. In most of everyday life you can get away with a great deal of imprecision in speech: your listeners will generally understand what you meant, and if they don’t, they can ask for clarification. In writing mathematics this is not the case: technical terms and notations have precise meanings and must be used accordingly. Unfortunately, this is a problem with no easy or quick solution.

The first step is to get the students to be able to recognize when they’ve written something that doesn’t make sense. In my experience the most effective technique is to sit down with a student and go over an exam or homework exercise in detail. In the example that you gave, for instance, you might ask the student what it means for $w$ to be countable. You may have to go back and forth with the student a bit to get a coherent statement, but eventually you should either discover that the student is a bit confused about the concept of countability or elicit a more or less correct statement. In the former case you try to determine just where the confusion lies. In the latter you might ask the student what the elements of $w$ are that are being matched up with natural numbers; with any luck the student can fairly quickly be brought to realize that the statement is nonsensical. Then you try to find out what the student was actually trying to say and show how it could have been said correctly. (Whether it’s actually correct in the context of the argument is a separate issue and should be treated as such.)

I would strongly suggest to them that whenever they use a technical term, they should consciously remind themselves of its exact meaning and verify that they’ve used it in a way that makes sense. I’d acknowledge that it’s time-consuming and that it’s very easy to let things slip by, and I’d explain that the goal is to reach a point at which one doesn’t have to think consciously about the proper use of common terminology and notation.

You might also point out that besides using language more precisely than we use it in everyday life, mathematics uses it a bit differently. In this connection some of your students may be helped by A Handbook of Mathematical Discourse, by Charles Wells; it’s freely available here as a PDF. His site abstractmath.org may also be helpful, especially the section Proofs and its subsection Presentation of Proofs.

Two last comments:

  1. Your task is a difficult one, and one that’s often quite frustrating; recognize that you’re likely to have only limited success, and take pleasure in whatever successes you have.

  2. The more time you can spend working with them individually, or at least discussing their specific errors, the more effective you’re likely to be.

  • $\begingroup$ I just took a quick look ofthe website, and I loved the Abstraction part. The main thing that worries me, is that they never get excited by what they are doing, since the class has got very formal, and them not having a good time with formality, maybe they can't even figure outwhat it's being done. In (my) class whenever someone writes something on the board, either them or me, I always ask them "why can we do this?" and "why can't we do this other thing?", I hoped questions like that would motivate them with the usage of the language, definitions, and the formality, but maybe it's not enough. $\endgroup$ – Ana Galois Oct 3 '13 at 12:28
  • $\begingroup$ (I'm painfully aware that I can only do so much until the semester is over and they keep actually trying, but I don't want to think that I didn't motivate them enough to get excited or interested enough by this, specially calculus, so they can countinue on their own). $\endgroup$ – Ana Galois Oct 3 '13 at 12:33
  • $\begingroup$ @Ana: It sounds to me as if you’re doing what I think are the right sorts of things. Sadly, even the right things don’t always work. These may be students who liked and did well at computational mathematics, but who just don’t ‘get’ theory. Or they might get more excited about some other flavor of mathematical theory; I’ve known students who did adequately at elementary number theory, for instance, but who had all kinds of trouble with elementary analysis, partly because they didn’t like that kind of mathematics as well, and partly because it really is a bit harder. $\endgroup$ – Brian M. Scott Oct 3 '13 at 22:40

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