Today I was looking through my introductory calculus textbook* for no particular reason. Well, I say introductory, but I think that’s a particularly bad choice of word.
You see, it was clearly intended to be an introductory text, but it failed at that rather miserably. I’ll give an example. Here’s how the textbook introduces the basic technique used to find derivatives (derivatives can give you the slope of a graphed function at any point on the curve):
To find the tangent to a curve $latex y = f(x)$ at a point $latex P(a,f(a))$ we use the same dynamic procedure. We calculate the slope of the secant line through $latex P$ and a point $latex Q(a + h,f(a+h))$. We then investigate the limit of the slope as $latex h to 0$. If the limit exists, it is the slope of the curve at $latex P$ and we define the tangent at $latex P$ to be the line through $latex P$ having this slope.
Whew. To figure out what that means, even to someone good at math, takes several moments of thinking to understand what the hell all the symbols and points and stuff are referring to, and how that gives the slope of a line. Compare the method used by the textbook to how Dave explained derivatives in his calculus tutorial (later edited and reposted by me). Sure, it’s longer Dave’s way, but you’re left actually knowing what is going on.
The textbook gets worse from there. It’d be more useful to someone who already understands the concepts and just wants to check some obscure property of logarithms or something. Understanding is buried beneath mounds of mathematical rigor.
If we want people to understand math, or at least not hate learning it, we’re going to have to make our textbooks less painful for a start. Take a look at the way Randall Munroe explains some basic physics in his blog: cartoon diagrams and jokes about death rays. Isn’t it so much more fun that way?
If I had more time on my hands, and if I could draw, I’d be writing a complete introduction to calculus. With stick figures, lasers, and actually understandable text.
Maybe I should try.
* Calculus: Graphical, Numerical, Algebraic, by Finney, Demana, Waits, and Kennedy.