In my (non-AP) calculus class last week, I was teaching my students about continuity of a function. Before we started, I asked them what continuity was, and students in both sections started their answer by saying “well, it’s when you draw a function and you don’t have to lift your pencil.” Some spoke of holes and asymptotes. Others spoke of endpoints.
I then proceeded to wow them by saying — all that they said could be encapsulated mathematically. The act of tracing out a function knowing where you’re going to have to lift it can be rewritten with three rules. They weren’t as impressed with that fact as I was, but I still tried to convey “Think about it! You can translate moving your hand across a page smoothly into mathematical statements.”
What’s needed for continuity of a function at :
1. is defined
I did the most obvious *you need to memorize this for tomorrow* wink-wink nudge-nudge that I possibly could. I might have even *coughed* the words “pop quiz.”
I just graded the quizzes. Horrible. HORRIBLE.
I got things that show no understanding of the symbols of calculus or what continuity means. Some examples:
(a) function exists
(d) one value for
(e) the two-sided limit of exists
(f) the two-sided limit of is equal to
(g) the function has to be continuous (you cannot pick up the pencil)
There are some major notational misunderstandings, but also part and parcel, some conceptual misunderstandings. I mean, for example, “the two-sided limit of exists” doesn’t really mean anything useful to us. First of all, it should be the limit of the function, and second of all, it doesn’t say the limit as approaches something.
I typed a bunch of these out and we’re going to talk about them in class tomorrow. Hopefully we’ll get to parlay that into a discussion of notation, the precise meaning of math symbols, and the importance of listening to Mr. Shah’s coughs.