Are questions on overlapping solids of revolutions without prior definitions and instructions fair given that there are divided interpretations?

Question

If words of command are not clear and distinct, if orders are not thoroughly understood, the general is to blame. But if his orders are clear, and the soldiers nevertheless disobey, then it is the fault of their officers. - Sun Tzu, The Art of War

Situation:

I am helping a calculus student, and it seems the professor of the class has given a homework item on computing the volume and surface area of a solid revolution obtained by rotating a region about an axis that cuts the region's interior such that Pappus's centroid theorems don't apply. There are no prior examples in class discussions or class tutorials, and both the student and I are unable to find examples in Calculus by James Stewart or the student's textbook, Calculus by George B. Thomas.

The student has been told that a classmate has asked the professor about this, and the professor responded, 'There is nothing wrong with that item.'

Apparently, examples of these are very difficult to find, and according to the answer given in the question linked above,

'There is not an unanimous interpretation of what is the rotation solid in this case.'

My questions

1. Is this fair to give this to students without prior definitions AND without prior instructions that it is up to the students to research the definitions?

2. Where can I read about this both for maths and maths education?

   • For elementary maths: What elementary textbooks give definitions for situations like these?

   • For advanced maths: What advanced textbooks discuss geometric, algebraic and physical interpretations for these?

   • For maths education: What does the theory of teaching solids of revolution to beginning calculus students have to say in this case?

I should note

This isn't the first time the student has encountered a homework item without a definition.

• In the semester prior to this, students were expected to compute the complex exponential $e^z$ in a homework item even though they had only the definition of $e^{iy}$, that is they were not told that $e^{x+iy} := e^x e^{iy}$. The student e-mailed the professor, who replied that the student was right that the students were not given this definition. This may seem like a small thing, but mistakes like this could lead to fallacies like because '$e^{iz} = \cos z + i \sin z$, $Im(e^{iz}) = \sin z$', in my opinion. (Obviously, complex analysis assumes calculus among others, so the class prior taught only complex variables, along with some probability and linear algebra, which had a little calculus but only up to what they knew in secondary school)

• The class also has an unresolved issue about critical or inflection points, though the calculus student in question has yet to contact the professor, and they are on break for the upcoming weeek.

Answer 1

I take it from the content that this student is at university level. Based on that, I'd say:

Yes, it is fair to give questions without having given examples before.

University is higher education. A key point of higher education, as opposed to further education, is learning to think for yourself. You don't learn to do that by never being given anything to think through for yourself.

The student does not need the definition $$e^{x+iy}=e^x e^{iy}$$ because they have already learned, years earlier, that $$x^{a+b}=x^a x^b.$$

On the volume of revolution: there may not be one single definition in use in the world, but there will be one definition in use in the class. The student should work from the definition they have been given. If theorems don't apply, then work from the definition from first principles instead.

Employers are not looking for graduates who can only follow previously defined algorithms. They can get robots to do that. Or high school leavers. They want graduates who meet a situation they haven't been programmed to deal with and at least try to find a solution.

Answer 2

1. Boy that's a leading question: "given that there are divided interpretations?" Seems like you are fishing for agreement or endorsement versus asking a question.

2. My practical advice is to just help out versus second guessing the teach. Even if the teach is actually wrong, then just say you can't help on that problem and move to a next one. But you're not helping anything with this "I'm a TA who disagrees with the professor" (and BOY, do we hear a lot of these types of questions).

3. For that matter, why not go talk to the prof yourself? Why come running to the damned Internet? Just engage in real life. (Even if you are not an assigned TA, but a tutor...then you have to realize there will come times it is difficult to be a tutor, given specific teachers.)

4. Also, you don't give us enough info about the specific homework problem (what object, what rotation, etc.) to see if the question is ambiguous or just difficult. Or even how difficult--maybe it's not as hard as you think.

5. The last example just seems like more of a rant about the professor. If you are going to tutor people and not engage directly with the professors, you are going to have these issues. You don't know everything that was said in class, for example. Doesn't mean you can't tutor. Just that you need to realize that it's not a perfect process.