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.