There are some very thought-provoking questions that can arise while discussing mathematical ideas with students.  Some intriguing questions that have emerged at different times (some repeatedly) in my classroom include:
   Why are there 360 degrees in a full circle?
   Why are there 60 minutes in an hour (or 60 seconds in a minute)?
   Why is our number system based on the number 10 and not some other number?

All very interesting ... but how do you deal with them during class?