I love when you are teaching and the students are not getting it (okay, THAT'S not the part I love) and then through some encounter with a student you realize what their misconception is. ... Then you totally revamp how you are presenting the concept, and they then all say, "Ohhhhhhh!".
For example, we (they) are learning sequences and have to deal with subscripted variables: an, an+1, an-1 ... I kept mentioning that the "n's" just indicate where you are in the sequence, but once they tried to apply it, they'd mess up. So .... "n" is the place you are (1st, 2nd, 3rd, ...) and "an" is the number in that place.
Well, I finally saw today that a student was mixing up things like an+1 and an +1 **, and then worked around to clear up THAT with the class. Ohhhhhhhh.
** let's say your sequence is:
Then a1 = 5, a2 = 10, if n = 4 then an = 20 and an-1 = 15 and an-1 = 19.