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I am teaching Calculus III this semester and a student signed up for this course after completed Calculus I and II in a different institution.

I quickly realised that this student does not understand many basic concepts from Calculus I and II. For example, she has never seen Polar Coordinates, does not understand Taylor's series, and cannot compute some simple integrals such as $\displaystyle\int e^{-x^2}x\mathrm{d}x$.

If she does not care about her grade in Calculus III and does not study for it, I probably will just give her an "F" and move on. But the reality is, she admits that she had a horrible Calculus teacher before and she did not learn much in her Calculus I and II class, but she got passing grades anyway and now she is in my Calculus III class. She works very hard and keeps talking with me and e-mailing me about her questions and difficulties. While I should be helpful to my students and I am trying to be, I also feel I am spending too much of my time on this single student.

When she seeks my help with concepts from Calculus I and II, I feel frustated as it seems I am paying the price for the incompetence of her previous calculus teacher. Should I tell her to retake, or at least sit in some Calculus I and II class? In general, is it a teacher's resonsibility to help a student with his/her problem from a prerequisite course?

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    $\begingroup$ Decide how much time you are comfortable spending and set the limits kindly. Brainstorm with her how she can learn these things on her own. I'm not a fan of Khan Academy, but many of my students are. She might hit that up. Do you have a list of Calc I and II skills she ought to have that she could use to catch herself up? I help students as much as I can. But I haven't had one asking this much in a while. I would probably give her up to 3 45 minute sessions a week, at most. And only if I felt like she was really benefiting. If your limit is less, that's ok. $\endgroup$
    – Sue VanHattum
    Oct 29 '20 at 2:44
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    $\begingroup$ As to the potential incompetence of the previous instructor: I think one contributing factor may be that some classes in the Spring didn't, in fact, cover all of their material. I usually don't fall for the "my last teacher didn't teach us..." thing, but in this case it may actually be the case? $\endgroup$
    – Nick C
    Oct 29 '20 at 2:55
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    $\begingroup$ Related from SE Academia: academia.stackexchange.com/questions/131849/… $\endgroup$ Oct 29 '20 at 3:49
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    $\begingroup$ And also: academia.stackexchange.com/questions/80524/… $\endgroup$ Oct 29 '20 at 3:49
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    $\begingroup$ Consider referring the student to some other support site at your institution: math workshop, in-house tutorial services, etc. $\endgroup$ Oct 29 '20 at 3:50
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It surprises me that you only have one student with this problem. I suppose you must teach at a school with highly selective admissions and high academic standards. I teach at a community college, and I would say that among our students who have taken first-year calculus, if the worst problems a student has are the ones you describe, then that student would be in the top 25%. Roughly half of our students have much more serious problems, e.g., they aren't clear on the difference between an integral and a derivative, or they notate the derivative of $x^2$ as $x^2 d/dx$.

It's good that your student understands realistically that she is unprepared. It's your job to help her, but the help should be helping her to plan a crash program of self-study to learn the material she didn't learn before. I would send her an email outlining to her what this would look like: --

Allocate about 2 hours per day, 7 days a week, to this for the rest of the semester. Go through your first-year calculus textbook and figure out which chapters you don't know. For each chapter, read it, then go back through and read it again, typing up notes on a computer (so that they're editable). These should be about half a page per chapter. Try to work the problems at the end of the chapter. Pick problems for which you have a way to check your own answer. Start with the easiest ones, and work your way up to the hardest ones. As your understanding improves, update your notes. If you get stuck, use the resources our campus provides. One of those resources is me, but I can't do the bulk of this, so you should mainly be using other resources: TA, campus math tutoring center, etc.

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    $\begingroup$ Per the last paragraph to the student: This is certainly the perfect advice to give a capable and self-motivated student, but I think making it an "assignment" would help keep them accountable, offering to have once-per-week check-ins if they want? "Try to work the problems at the end of the chapter. Pick problems for which you have a way to check your own answer." I would just give them the list of problems I usually assign in that class so they're not staring down the barrel of 85 problems per section, probably not aware of which ones are vital. Do you think that is giving too much? $\endgroup$
    – Nick C
    Oct 29 '20 at 14:03
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    $\begingroup$ @NickC: Those seem like possible reasonable things to do, if they're willing to devote this much time to it. In my experience, the reality is that this kind of self-study plan has maybe an 80% failure rate with my students, because they're simply not sufficiently mature and self-disciplined. For many students, 14 hours a week would be several times greater than the total time they expect to spend on all their courses. If they were that mature and self-disciplined, they'd probably have learned the subject the first time around, regardless of the teacher's standards and quality of instruction. $\endgroup$
    – user507
    Oct 29 '20 at 14:25
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    $\begingroup$ Thank you so much!! I have just e-mailed her your last paragraph (after some minor editing) and it should be very helpful to her. $\endgroup$
    – Zuriel
    Oct 29 '20 at 15:23
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Here is some complementary advice to that given by Ben Crowell.

To prioritize the remediation effort, my advice to the student would be to start with the notes she takes in your lecture. Notes taken on the example problems should include both the formulaic steps of the problem but also some written terminology about the technique, especially if the justification for the manipulation is unclear. Then, after the class, she can review the notes, use the terminology to search in the index for a section which explains the technique, and proceed as Ben suggested. If the section does not make sense, then go backward one section at a time until the discussion is comprehensible. After that's done, she could take a stab at the homework and follow up with you or another resource (TA, tutoring center, supplemental instruction, etc.) for an explanation to drive more index-driven research.

This approach balances the need to re-learn the old material with the reality that your course is still moving along and takes advantage of work that's already being done (sitting in your lectures, taking notes, doing homework) to drive the prioritization of an otherwise daunting task.

You may be doing this already, but one thing you could do to facilitate this would be to just mention the name of an old technique even if you do not write out all the details (e.g., "we evaluate this integral using trig substitution" and then just show the final answer). Name dropping the technique probably would not take much effort but would give confused students something to look up in the index.

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To answer the title, I'd say you're not responsible since Calc I and II should be prerequisites for Calc III and so it's expected that she know that material. Not sure how she has never seen Polar or Taylor (sounds like her prior classes didn't finish in time?)

I'd probably say "F" and retake, and/or sit in Calc I and II classes. The reason I say this is because I only foresee this getting worse. If she doesn't have the foundations of single-variable, then a lot of the integration and differentiation that is taken for granted in going to be too much. What happens when you get to cylindrical and spherical Coordinates, or double integrals, change of variables, etc.?

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