Hi. Occasionally a student will ask me a question, or will make a mistake, that leads to a topic that is beyond the scope of the course. Recently this occurred in college algebra where students confused algebraic functions and algebraic numbers. This entire subject is beyond the scope of this course - we don't prove a function is not algebraic, but we merely state it without proof; and we really don't care for this course whether a number is algebraic or not.
My philosophy is that I always try to answer a student's question, or try to correct a mistake, even though this gets us outside the scope of the course. Students will not be tested on whether a particular function is algebraic or not. But I always have this nagging feeling that I'm providing more information than anyone really cares about.
Thinking about this dates back to when one of my sons was in middle school and says he was taught that only rational numbers can be exponents. I then dashed off a letter to my son's teacher, reminding him that while logarithms were not part of the middle school curriculum, did he really want to teach the kids something that they would have to un-learn in a future class? I try not to teach something that a future teacher may have to correct; my son's teacher felt an irrational exponent was too much information for that grade level.
Which of us is right?