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I will be working as a TA in Calculus 1 this semester. In Norway, this means leading a study group with about 20 people (most of the days, I guess around five or six will show up), writing up the exercise-numbers on the blackboard, and answer any questions they have about them. More than likely, I will get students without an interest in mathematics (biology students, chemistry, geology etc.)

If the students have no interest in learning the rigorous and (subjectively) interesting parts of the subject, should I still force these concepts upon them? Many of them will be forced into taking more advanced subjects later (Linear Algebra, Calc 2). However, the nature of Calc 1 allows you to get a C, or maybe even a B, by simply cramming methods. I am having a hard time making up my mind about this, because it is very likely that my students wishes (and maybe what's best for them in terms of grades) will be in strict conflict with my philosophy as someone who hopes to become a good expositor of mathematics.

(The best example may be the formal definition of the limit. This can be taught well, as in, "this is why it works", or by simply reducing it to a humanized algorithm, which will often give you a correct value of delta.)

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  • $\begingroup$ Can you explain what you mean by "force it upon them?" The title of your question implies that it is possible to give someone an understanding, which is misleading. I am guessing that you are asking whether you should give more advanced assignments, or require certain kinds of responses that are not expected in the class they are taking. But it would be better if we could understand specifically, and practically, what it is you are suggesting you might do differently in the two cases. $\endgroup$ – JPBurke Jul 25 '14 at 21:30
  • $\begingroup$ An example would be a student getting a wrong delta while proving a limit. I may simply point out the error, ignoring the student's lack of understanding of the method, or give the student an explanation of how the method actually works, which may or may not allow the student to find such errors more easily themselves. $\endgroup$ – Andrew Thompson Jul 25 '14 at 21:52
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    $\begingroup$ You never know what showing genuine interest in math will do, some students have never been exposed to someone who wants to really understand math as opposed to just "do it". At least, in my experience, there are occasions where exposing non-math students to real math may turn them into math students. That said, you're a TA so don't get too carried away in the required component, unless your professor is on the same page. $\endgroup$ – James S. Cook Jul 26 '14 at 0:52
  • $\begingroup$ However, the nature of Calc 1 allows you to get a C, or maybe even a B, by simply cramming methods. I'm not sure what you mean by "cramming methods." I would hope that a biology major could get an A in calculus by mastering the methods and understanding how to apply them. Newton and Leibniz went to their graves not knowing about limits, and I would hope they'd get A's in calculus. $\endgroup$ – Ben Crowell Jul 26 '14 at 1:24
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This is a very narrow answer, but I feel that if you construct the questions cleverly, you can teach the neat concepts conceptually, even to previously "uninterested" students.

For epsilon-delta, I've had some success with questions like:

  • Suppose someone is trying to prove that $\lim_{x \to 3} x^2 = 12$, and tries $\epsilon = 8$. Find a value of $\delta$ that works with this $\epsilon$.
  • What is an example of an $\epsilon$ that will not work when trying to show that $\lim_{x \to 3} x^2 = 12$?

They almost can't help but gain some understanding of why it works if you ask this kind of question.

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  • $\begingroup$ [possible answers are $\delta = 1$ and $\epsilon = 2$, respectively.] $\endgroup$ – Chris Cunningham Jul 26 '14 at 5:43
  • $\begingroup$ Just to follow up, I'm starting to find that intro calculus students can completely grasp this kind of question, but have trouble connecting it to the "proof" that we are familiar with as mathematicians. I also suspect that the mechanical operation of the "proof" is kind of useless, and I might transition to covering epsilon-delta primarily through these kinds of questions. $\endgroup$ – Chris Cunningham Feb 24 '15 at 20:47
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I imagine a scenario like this: you see an error and ask them how they arrived at what they have written. They can't explain it. You then ask them whether they want an explanation.

I suppose a student might say "just tell me the answer" but that's treating you like a glorified answer key. How will they get the answer on the class' assessment? But their reasoning may be different, even if they did decide, in some way, to come to a tutor.

I think your focus should be on understanding. But, then, that's my philosophy. When you're a tutor, you're working for someone and they may have different ideas. Your colleagues can support you in your philosophy, but some people are hired (for example) to do test-prep work and would not get rehired if they did not use cramming approaches that are (let's be honest) near useless for understanding.

While someone else may have an insight from their experience that will help you, I'm going to say that this may be very much a personal decision because of the employment ramifications. If I am understanding correctly.

A somewhat analogous situation from public school: a school administration in public school may mandate a certain curriculum pr pedagogy. Teachers very often face philosophical differences. They may well ask "should I follow what my math coordinator and district want me to do, or should I go outside of the acceptable leeway I've been given?" If one is in a situation with personal consequences, then it's a personal decision. Especially after other measures (like petitioning the administration) have already been tried.

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