It is accessible to a high school student. Yet has insights for a PhD in mathematics. And more than any other book I know, it captures the experience of mathematics. Both good and bad.
I like to say that I went into mathematics for reasons given in that book. And I left it for reasons given in that book.
The analogy is to teaching algebra through endless, dry repetition of problem solving. Some of you won't get what the big deal is; you're the ones who are good at 'crossword puzzles'.
He taught math by talking about the history of the problems we were trying to solve, and how people went about solving them. It was the concepts that mattered; the equations were secondary. He would put a problem on the board, and ask the class to solve it. He would then give us pieces of the puzzle, until we had enough that we could figure it out ourselves. I found the class participation to be key. With most math classes in college, the instructor works through 3 or 4 problems for a given topic on the board, and the students spend the entire class furiously copying down what he writes. It's pointless. I get that it's driven by the need to cover a lot of material, but I'd rather the class work through two examples then the instructor work through 4. Also, regular quizzes were a key part of Irani's class that made retention much, much easier. Instead of the typical weekly homework, two midterms and a final, I would much prefer weekly quizzes, 3 (or 4) midterms, and a final worth a much smaller chunk of the grade.
Last semester, I took linear algebra. We had weekly assignments that I did with a tutor (bad decision), two midterms, and a final worth 45%. I went into the final with ~40% in the class, crammed for it, and ended up with a mid-60 in the course. It was pointless, and I don't feel like I learned much. I would much prefer the system Irani set up, but he's not a university professor, and he doesn't have a PhD. He taught himself calculus with an inch-thick pocket book he picked up from a second hand store.
Feynman on magnetism: "I can't explain that attraction in any of the terms that's familiar to you. For example, if we said that magnets attract as if they were connected by rubber bands, I would be cheating you[...] and if you were curious enough, you would ask me why rubber bands tend to pull back together again and I would end up having to explain that in terms of electrical forces."
Math education today is broken.
UPDATE: Link to talk with some other resources: http://computerbasedmath.org/
Thanks to Khan Academy, I now understand trigonometry in a much cooler and intuitive way.
However, I don't know if that mean I am beginning to transcend the mechanics/grammar of mathematics.
Then you take foundation of mathematics, metamathematics and more of that ilk and you are labeled a Platonist. Well, guilty as charged.
Writing software doesn't help you discover the laws that govern everything, but it affects people on a much more pragmatic level - software satisfies our curiosity, tells us how to go where we need to go, and affects every aspect of my day-to-day life. And that's not half bad.
In retrospect, I should have been a coder. Although all my coding adventures post high-school have basically been autodidactic, I jumped in the deep end and took a high-level undergrad/grad CS class and coded my own web server... I was the only person whose server didn't allow the user to drill through the directory structure and hack the server because at the time I didn't know the posix file functions and actually wrote my own string parser/handler.
She attributed that not only to the fact that he was very smart, but also that he had mastered the Vedic techniques that let him quickly organize, simplify, and compute long strings of numbers and operations in his head.
I've never heard anyone call Gymnastics "Gyms". :P
On a grander scheme, how can math describe things like love, goodness and beauty?