Now that I'm 3/4 of the way through writing the Intro to Geometry textbook, I've finally started doing some serious thinking about what I think makes a good math text. I was prompted to think about this in more detail recently when I realized that it's the absence of good math texts that is in part responsible for the fact that I stopped independently studying higher mathematics. (There are a stack of other reasons I won't go into, but if there were excellent teaching texts, I would probably still be learning math outside the math I learn through AoPS.) Later, I'll come back to what I'd like to see in a textbook, but at this point I'll focus on what I think is wrong with textbooks - this is what inspired us to write the original AoPS, and what inspires us to write the new AoPS series.

Simply put, most advanced texts are not written for people to learn from. Non-advanced books aren't any better, as usually they offer little more than simple recipes, with no deep appreciation of the subject offered. I think the fatal flaw many of the authors made when writing advanced books is striving to present the mathematics in a clean, professional way. That probably sounds like something to strive for, but it doesn't work in a math text. It's perfect for a reference book, but for an instrument of learning, it's usually a disaster. The fatal flaw in the basic books (e.g. middle and high school texts) is that they are written for a market that has warped math education into its current form. I'll focus mainly on my complaints about advanced books, since the shortcomings of standard middle and high school texts for eager students are, well, self-evident to most math lovers. It's less obvious to a mathematician unaccustomed with education why the advanced books are not effective.

First, the books are written in the style of a professional mathematician. The trouble here is, if the book is meant to be used for learning, as opposed to for reference, then the target audience should be people who are NOT professional mathematicians. The style should reflect the reader's level of understanding mathematics, not the authors'. And to the argument 'they have to learn to read professional mathematics', I counter that there are far better places to do that - readers are struggling enough to grasp the material, why complicate it by making the language and style foreign, as well. Furthermore, the argument that 'they have to learn to read professional mathematics' is somewhat akin to just giving them the book in plain (un-compiled) LaTeX and saying 'well, they have to learn how to deal with LaTeX'. Absurd. Teach one thing at a time. Teach math with the textbook, teach style with professionally written mathematics on subjects the students already know.

Next, the '60 pages of vocabulary followed by 150 pages of the theorems and corollaries' structure that many books follow is excruciating. The 60 pages of vocabulary is nearly impossible to wade through - I have several books in number theory, graph theory, algebra, etc., that I would love to learn, but there's not a single interesting idea in the first 30 pages; however,I can't understand the interesting ideas once they're presented without the vocabulary. It doesn’t have to be this way - maybe I can't do the high powered stuff without the vocabulary, but throw the reader the bone - give the reader something interesting to think about. I refuse to believe there's no way to engage the mind without first defining everything in sight.

Then there's the 'theorem and corollary' structure. This is entirely encyclopedic - here's the info. It's not instructional. There's no motivation for why we think of these things (and I'm not talking about the standard US textbook with their so-called 'real world' problems, a trap I fear MATHCOUNTS may have fallen into, but that their skilled question writers might dig them out of yet - a rant for another day). I'm talking about inspiring the readers to develop the ideas on their own. Sure - you need a trail of breadcrumbs for the reader to follow, but make them walk the trail.

But when they walk the trail, provide some reinforcement - textbooks without solution guides are poison for self-study. (As an aside, if I had to name one failing of Zeitz's Art and Craft of Problem Solving, a book I highly respect and use in the Independent Study, it would be this - virtually no solutions.) Rigorous, efficient self-study should include some sense of discipline for 'time to give up on this problem, read the solution and learn from it and move on' (and perhaps revisit the problem in a week or two to confirm I've internalized how to find the solution). This is very difficult, if not impossible, without solutions.

I'm not sure what inspires these shortcomings in most books. I think the style and structure issue are a result of people too far removed from the learning process themselves. (It may also be a demand of the marketplace - the people making decisions about buying a book to be used as a text in a class expect an encyclopedic book they appreciate, as opposed to an effective teaching tool.) As for the solutions issue, skipping that has a clear motivation - they're a grind to write!

However, I think at heart, the problem with the books is that the writer does not understand the audience. A perfect example of this is Feynman's Physics Lectures. They're brilliant, and I love them. They were largely a result of his teaching the intro physics at Caltech. They were an utter failure then, as I understand, and the Lectures make clear why. They are brilliant and wonderful, yet completely useless for anyone who has not already internalized the ideas of physics to some degree. This explains why grad students and young professors would attend the lectures by the end of the semester, but most of the new Caltech undergrad students had already bailed. This last bit may be legend, but it doesn't change the fact that as a pure teaching tool to new students of physics, the Lectures are a failure. However, as a book written for a person already conversant with physics and math, they are a gem.

This, then, I would ask of text authors - know who your audience is, and write the book they want. I hope we are doing this in our new series. Our audience is a student who wants to learn math but doesn't already have some degree of experience in whatever it is we're writing about. I'll write more later about what I think an ideal textbook should be, and how I hope we're reaching it with the new series.