Cap’n Refsmmat’s Blog of Doom

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 at a point we use the same dynamic procedure. We calculate the slope of the secant line through and a point . We then investigate the limit of the slope as . If the limit exists, it is the slope of the curve at and we define the tangent at to be the line through 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.

A long long time ago I discovered Project Euler, a competition of sorts where the goal is to solve numerous mathematical puzzles with the aid of a computer. I originally thought it to be too difficult, but recently I revisited the site and discovered that I could actually solve the problems with what I’ve learned over the past few years.

Those of you with a programming inclination would probably be interested: it requires the use of some very clever techniques (so you can calculate such beastly numbers as 4000 factorial relatively quickly), and best of all, you can use any programming language you’d like.

So far I’ve solved 21 of the problems. I’ve mostly been brute-forcing the solutions (”guess and check”), but as I work my way into more complex problems which would take hours to brute-force, I’m being forced into cleverer and cleverer programs. It’s very intellectually stimulating.

If you’re up for it, try it out with your programming language of choice. It’ll help both your math skills and your programming skills — there’s nothing like trying to find the best way of calculating 4000 factorial in Lisp to make you learn about the language.