In my 9AM calculus class, students don’t have desks, but have tables where they sit in sets of four. I like this setup because it makes it easy to assign group work and the students already have a pre-made small group for them to work with.

One of the pods has a bunch of guys who seem to not care about the course material. They talk with each other while I’m lecturing, crack jokes, and can be a bit annoying at times. But they ask GREAT questions, and the best part is they aren’t aware of it. One such question came after I reminded the class for the umpteenth time to make sure to read through the section we’ll be covering both before class and after class. One of the students in that pod asked a question which was surely meant to get me off topic that way we wouldn’t have to do any more work on limits.

So how do I read a math text? I read it a lot, like over and over and over again. In my first read of a math text, I look for the story of the section. There is a reason that all of these topics are grouped into one section, and the goal of my first couple read-throughs is finding out that reason. I look for the overarching story which connects those topics. If I run into a result I don’t quite understand, that is totally fine. I mark it as something to return to and keep on reading. The focus of the first (or even second) read is finding out that story. It’s only after you have that story that you can start to graft on the relevant details to flesh out the theory.

In the next section of the book (Sewart’s Calculus: Concepts and Contexts, section 2.6) we’ll be first talking about the derivative. So how would an ideal student want to read this section to prepare for class?

The first thing that the section talks about is the tangent line. It defines the tangent line to a curve $y = f(x)$ at the point $P(a, f(a))$ as the line which passes through the point $P$ with slope $m = \lim_{x \rightarrow a} \frac{f(x) - f(a)}{x-a}$,

basically the slope is the limit of the average rate of change over the interval $[x, a]$. The student has already seen many examples of looking at the average rate of change over smaller and smaller intervals, so this should look relatively familiar to the students (if they’ve been diligent with their reading, homework, and group work up until now–I understand that that may be a big assumption for some students). In the rest of the section, the exact same process is done twice more, once in the context of displacement functions and once where he introduces $\frac{\Delta f}{\Delta x}$. This then brings up a question: Why are we doing the same thing three times? Thus ends the first two reads.

With another read or two, students can work through the examples and then make the connections between the three forms of the derivative (namely that they are all the same) and make sure that they understand the tools.

But the first step is to find the story. Forget the details; find the story and then the details will make more sense.