How to prevent too many students from making appointments to meet with you when you put "Office Hours: ..., or by appointment" on the syllabus? In the past I have included the following statement on my syllabus:


  Office Hours: Mon 1-2pm, Weds 3-5pm, or by appointment


However, I am beginning to get too many requests for appointments. Appointments are really inefficient. They waste both of our time trying to schedule them, and with multiple appointments per week in addition to office hours, even holding them takes a lot of time away from research. How do you accommodate students who can't make your office hours without devoting too much time to this task? 

I thought about setting up a class message board where they can ask each other and answer each other's questions - anonymously if they wish - (I can also answer questions on the message board) and put on the syllabus "All content related questions should be posed in class, on the message board or in office hours. I do not reply to questions about content via email. Please schedule an appointment to meet with me if you are having difficulty getting a question answered after trying the above options"  

Some Background: I'm a graduate student lecturer. I teach 2 sections of calculus, so about 60 students. Since classes are small (~30), there are about 10 lecturers (some are faculty and some are grad students with masters degrees) and we all have office hours. Mine tend to be very crowded, about 5-10 students showing up (some from other sections). I tell my students to go to other instructor's office hours, but they say they aren't as helpful.
What an interesting problem to have. When I saw the title, I thought, "This must be hypothetical, because this problem never, ever occurs." But I was wrong.

Anyway, I don't know how to deal with having too many students try to schedule office hours, because I have never had that problem, even when teaching 60 students. 

I do, however, employ the message board tactic that you mentioned, which probably helps cut down on office hours (and dramatically reduces the number of emails I get). I suggest the free (and extremely slick) service Piazza. Students can collaboratively (wiki-style) answer each others' questions, and instructors can edit everything in sight, as well as create their own answers. When posting questions, students can also appear anonymous to their classmates, but not to the instructor, which eliminates any embarrassment they might feel. 

tl;dr: Try Piazza.