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One of the most common replies I have gotten as a student in engineering is the phrase "what you are asking is beyond the scope of this course". But I always found it a bit funny coming from the prof since he or she most reasonable have a thing or two to say about the subject. Furthermore, whether something is considered "beyond" sometimes depends on a prof's temperament, on a good day a topic that is beyond will be addressed, or a busy day that topic is hold off indefinitely.

Now I am a TA for an introductory calculus class. Often there would be a handful of students come into the class with years worth of experience in calculus. Sometimes they will ask a question that is addressed in an upper year course, sometimes the question would have to resort to complex variables, sometimes it relates to physics.

How should I handle students who are interested but asks question beyond the course in the sense it requires an additional course or two to truly appreciate its importance or at least to see how the actual calculations are performed.

I could tell them the answer but sometimes it can lead a student down a rabbit hole which can be devastating given how busy first year students are.

Further, I don't want to disrupt their "natural course" by saying something that may prevent independent self discovery.

Lastly, I don't want to say something which could be misconstrued as a test topic.

At the end of the day, how should I address the questions that are deemed beyond the scope while not withhold information.

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"This is beyond the scope of the course" is not a great answer without further clarification or commentary. As you say, the ethos of the university is that your instructor is someone whose qualifications and expertise lie far beyond the scope of any undergraduate course. The answer is justified for a student who asks a certain kind of question in class, because class time is limited and one must exercise judgment about what to say and cover in that limited time. Going off on a lengthy digression that is likely to be of interest to only one student and perhaps not even well understood by her is not a good use of class time. So I would expect an instructor to say "Come talk to me after class if you are interested in that."

If a student comes to talk to you in your office hours or your spare time, I think that she deserves some kind of answer. The answer may in fact be that the question lies beyond your expertise (and there is nothing inherently wrong with that; there is a lot of stuff out there...), but in that case you should still spend at least a little while trying to direct the student elsewhere, either to the appropriate reading materials or to some other faculty member who can better help them out.

If you do feel that you know the answer to the question -- or at least, enough of the answer to the question -- then, sure, take a crack at answering it. It takes a lot of expertise -- subject expertise, pedagogical expertise, and practice -- to be able to give answers to such questions which occupy a reasonable amount of time and are at least somewhat meaningful to the student. This may involve for instance asking some quick questions of your own, trying to understand the student's background and the true direction and depth of their interest. One mistake that even seasoned pros make is to open up the gates and flood the student with information of a quantity, density and sophistication that is beyond what they can be expected to process in the moment. If someone asks you about the example of a conservative vector field on the punctured plane which is not a gradient field that you discussed in class, you should probably not respond by giving them a half hour lecture on DeRham cohomology. (At least not at first. One of the amazing things about teaching is that the chance that a multivariable calculus student really is looking for a lecture on DeRham cohomology in answer to their question is very, very small...but it is positive!)

How should I handle students who are interested but asks question beyond the course in the sense it requires an additional course or two to truly appreciate its importance or at least to see how the actual calculations are performed.

You take a shot at it. Make your first shot very brief: just drop some terminology and try to give a sentence or two expressing one of the main ideas in broadest terms. In the above case, you might say "Whether every conservative field is a gradient field depends on the domain. We saw that this is the case when the vector field is defined on the entire plane [or all of three-dimensional space...]. It is also true if the domain is something like an open disk or ball. It turns out though that 'holes in the domain' lead to conservative vector fields which are not gradient fields." (By the way, I first wrote more and then deleted some of it! Restraint is truly hard.)

I could tell them the answer but sometimes it can lead a student down a rabbit hole which can be devastating given how busy first year students are.

I'm not really sure what you mean by this. The type of personality that is going to be "devastated" by learning that things go deeper than they currently know does not seem well suited to higher education. If anything I feel exactly the opposite way: as an educator at any level, showing students the rabbit hole is one of the most important things that you can do. Especially, getting an undergraduate degree is all about learning just enough to get an awareness of the true depth of knowledge and acquiring a sound foundation upon which more knowledge can be built.

Further, I don't want to disrupt their "natural course" by saying something that may prevent independent self discovery.

Again, I don't really buy into this. Students who want to be sufficiently well insulated from being taught things by other people do not belong in a university. Self discovery is a wonderful and important thing, but it is enriched and reinforced by prior knowledge and coursework, not ruined by it. There are plenty of things to discover for oneself, and anyway you are only telling them a little. It seems very likely that such a conversation would if anything trigger the student's independent learning and self discovery, not inhibit it.

Lastly, I don't want to say something which could be misconstrued as a test topic.

This is why it's best to address such questions outside of the classroom, or at least outside of the class session, and ideally with only the students who are explicitly interested. It should then be much clearer that what you are telling them is not a test topic. If there is any ambiguity about that then you should resolve it. For instance, maybe the question actually is closely related to a test topic but comes from a direction in which the student does not see that. In that case you should point out the connection to them.

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    I think it is always worth at least mentioning in a sentence something like "Even though we have local potential functions, we cannot assemble these into a global potential function on the whole punctured plane. This is the beginning of a beautiful subject called Algebraic topology, which is concerned with such phenomena on more general spaces". – Steven Gubkin Mar 7 '15 at 3:27
  • Yeah, exactly. I usually quickly give an answer that gives the student some idea of what to do next and the potential complexity involved and then get right back to whatever it was I was talking about before. – Dave Kanter Jun 17 '15 at 19:51
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From a student perspective, a short answer that just mention the concepts I need to know to solve a certain problem I am interested in is pure gold. If it comes with a reference or two, it's even better.

I think most students (especially those who ask that kind of questions) are able to manage their time efficiently and decide what is more important for them. Also, some of them actually enjoy what they are doing (some ask just of curiosity) and want to study that topic more deeply, in their own spare time. From my experience, knowing what to look for or where to look for it saves a lot of time. Considering the fact that if I ask such a question, I'm usually really interested in that topic, I'll waste that time to find the concepts/references that I need.

So, for questions that are way beyond the scope of the course, I think the following answer is ideal:

This question is well beyond the scope of this course. In order to solve your problem, you require concepts A, B, C, which are studied in more detail in courses X, Y, Z. But, if you're really interested, you can find some information in the following textbooks ...

I don't believe that this is too much time wasted on your side, given the fact that you are aware of the required concepts. Also, it is not harmful to the student, as you make him aware that if he chooses to study this in more detail, he should do it on his spare time at this moment. Also, you inform him about a possible academic path, by giving information about which courses study the topic in more detail. And, if he really wants to study the topic in more detail, he has the references required and knows what to look for.

Also, it might helpful to point to another academic who is more entitled to give an answer like above.

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Write down the question: I always do this, even with questions I answer right away. It shows me the gaps in my teachings. By the students' questions I can adjust my material for the next time I teach a topic. Taking a note is also the basis for all other recommendations.

Recommend books: Tell the students where to find that topic treated. This also "forces" you to keep up to date about material that would else lie beyond the scope of your own preparations. It's never bad to know more than you have to teach. Since I have written down the question I can check what books are available and give more precise recommendations (e.g. "Chapter X in book Y treats this topic thoroughly - but I find the whole book very interesting.").

Office hours: Since I wrote down the question I can do some short research to be up-to-date when the student (or group of students) comes to seek my answer on it.

Breaks/After lecture: My least favourite way of handling this. First of all you're under time pressure during breaks and after lecture (cleaning up the space for the next lecturer, etc.) and secondly there's no time anymore to really have a look at material covering that topic. I prefer to be up-to-date before giving an answer.

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Excessive enthusiasm or curiosity is usually easily handled. Unless you have Feynman Jr.

Obviously the primary responsibility of the prof and TAs is to teach the on-course material, and teach it clearly and well. If not, you deservedly get into trouble. If you confuse some students or they mistakenly think something is on-course when it isn't, you could get into trouble. But, you also want to encourage curiosity and give students signposts to how everything all fits together and what future courses build on it, and how it relates to other disciplines. Yes, it's a judgment call and depends on your/professor's workload, mood and energy level that particular day.

So as to material beyond the scope of this course, IMO that falls into different categories:

  • a) stuff that's beyond the scope of this course, but will be covered in follow-on courses (core or popular electives).
  • b) ... stuff that's really advanced or only covered in rare electives.
  • c) ... stuff that's generally offtopic for most students, as in essentially irrelevant.
  • d) ... stuff that's generally offtopic for most students, as in is only covered by further degrees or diplomas, possibly in a different field, e.g. Masters in Quantitative Finance.

a), b) you try to encourage their curiosity without dedicating class time or injecting confusion about what's on-topic. c), d) you gently tell them is offtopic and point them to the library for any independent reading they care to do.

I guess your specific case is kind of the flipside of a): e) stuff that can be understood by people who have done other prerequisites or courses

I would briefly mention to them the advanced calculus or physics implications of something, while preambling "this is not on the course, just for the benefit of people who've already studied X,Y,Z". If the more basic students object, you may have to tone it down. Possibly better to give these pointers in writing (/on course website) so doesn't take up class time or confuse the others. "See me in office hours" might be one tactic but don't let it preempt teaching the essential material.

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