How to Explain Mathematics


Excerpt from a fan letter:

> Explanations of mathematical ideas in a way that makes
> them hard to NOT understand are extremely rare and I must
> thank you for making yours publicly available.


You know, there's a subversive intent in my web page articles, namely to change the way people teach mathematics by making much more use of specific examples instead of giving extremely formal presentations that cause students to get lose in a forest of symbols. I first started doing this because I was frequently teaching a course for elementary school teachers, many of whom had only a shaky grasp even of elementary algebra. I found that most of them were completely unable to deal with any equation that had more than two symbols in it. And while many of them might be able to understand an identity such as

   a2 - b2 = (a + b)(a - b) ,
and even use it in doing algebraic manipulations, they didn't understand what it was really saying in the same way they did if I showed them
49 - 16 = (7+4)(7-4),
     36 - 25 = (6+5)(6-5), etc.
This was the beginning of the idea for my Lazy Man articles.

And then once I saw how well this worked for my elementary school teachers, I started using the same sort of explanations with my calculus students.

In fact, this is exactly what I do myself, and I suspect everyone does, when reading mathematics. I read something that seems very complicated, and then I ask myself, ``Well, what would this mean in terms of a specific example?''

To the extent that I explain things well, though, a big part of it is that almost all my articles are a result of having taught the same material over and over again many many times. And one day, when teaching Green's Theorem, for instance, for the umpteenth time, I would stop and think to myself, "You know, I can certainly see why each of the steps in this proof is true, and I can certainly present them one more time in class, but I never have understood why the damned theorem is true." And I would then try and step back from the trees and see the forest, as it were, and figure out what the proof was actually saying and why anybody could have ever come up with the idea for it.

Also, I don't worry about being long-winded and redundant. I know that some people will be annoyed, but a lot of other students will need to be told the same thing several times in order to get it. And I let myself put in lots of white space. If it takes an extra page or two on somebody's laser printer, that's a small price to pay for making it easier to read.

The thing is, almost all mathematics textbooks are written by mathematicians, who know lots about mathematics but know absolutely nothing about how to write a how-to book. When I look at a book like HTML for Dummies, I think: Why can't a calculus book be written this way?