How to handle relevant, but disruptive, questions I've seen on this site several posts about how to handle silly or "trolling" questions, but I recently faced the opposite situation. Several students in a class I'm teaching continuously bring up questions that are relevant to the lecture and/or interesting; however, they ask so many questions that it tends to disrupt the lecture by throwing off the pace and flow of the presentation (e.g., I lose my train of thought or I have to speed up to finish the lecture on time). I don't want to discourage questions, but at the same time, I don't think it is fair to entertain so many questions if it results in a lowering of the lecture's quality. 

A solution that I tried is telling the students that they can come to office hours or make an appointment with me to talk. However, most of them don't take me up on those offers, and those who do take advantage of my offer sometimes come with so many questions that I cannot answer them all within the time I've set aside for office hours/teaching appointments.

Since this is limited to a small (but vocal) subset of my students, I don't think the issue is with the quality of my teaching. Otherwise I would expect (and hope) that more people would be asking questions or giving some other indication that I'm not being clear.

Is there any way to handle this sort of situation without coming across as uninterested in student questions or discouraging questions from being posed? I truly am interested in making sure everyone is understanding the material, but at the same time, I cannot be devoting large amounts of time above and beyond what I'm already devoting to teaching.
As a student, I find that waiting to ask all questions in office hours or the end of class is not a great solution. I don't mind delaying non-urgent questions, but sometimes if I don't understand something presented in the first ten minutes of the class, I might then miss out on the rest of the lecture if I don't have a chance to ask a question. So I appreciate when instructors give students some chance to ask questions during the lecture.

A good way to allow this and also control when and how many questions are asked is to invite questions in each class at times of your choosing (and let students know you'll be doing this).

At the beginning of the lecture, say, "I'll pause and ask for questions at regular intervals, so you'll have a chance to ask your questions then."

Decide ahead of time where you will pause and ask for questions (at convenient places to break the lecture, or places where you know students often have questions).

During the lecture, pause at the predetermined places and say something like:


  "I'll pause for questions now. We have time for three or four questions."


(and you can decide how many questions to take depending on how the lecture is going, time-wise.)

Students with too many questions in office hours are an entirely different issue. You should ask these students to schedule a separate appointment, and determine whether the questions are a sign of a problem (i.e. they may need tutoring or similar services) or a good thing (i.e. they are looking for enrichment and need some pointers on how to find it).