Earlier, I ranted about what I didn't like about most modern texts, and why I have more or less stopped reading them. Now I'd like to write a little more about what I think texts should include. Hopefully, we'll be able to do most of these things in our new series of books. I won't focus on the things I think obvious - a good index (hard to do), a complete solutions manual (mandatory here at AoPS), few typos, etc.

First, the language should speak to the reader, not be an example of how the reader should aspire to be when she is a professional. Ideally, this means a conversant tone - this book is meant to teach math; it's not the time for teaching professional mathematical communication. I think this is a large part of the success of the original AoPS books, and something I didn't appreciate at the time. Over the years, we've heard dozens, if not hundreds, of people say that the books are like having someone talking to you. I'll credit Sandor for this - early on he ruthlessly removed the stuffiness from my writing. I don't know if it was strategy on our parts, or simply the fact that we were 21, but our writing turned out a lot closer to how a 14-17 year old thinks than most textbooks do.

Second, the structure should engage the reader. I strongly believe that most people learn math by working problems. Presenting theorems to read doesn't effectively engage the reader - giving them a problem to try to solve does. Obviously, these problems must be chosen very carefully. Inspired by comments by Josh Zucker at a conference last December, we've taken this approach to a bit of an extreme in our new books - instead of teaching a lesson before presenting problems, we present the problems that inspire the lesson first, thus inviting the student to develop the lesson on his own. When you discover knowledge on your own, even if a trail is laid before you, you own it in a way you simply cannot if the knowledge is simply handed to you.

Third, the practice problems should follow a range of difficulty, and there should almost never be a reader that can do *every* problem on her first try. However, there shouldn't be so many problems that a student must make her own call about when to move on - there should be just enough at a given difficulty then move on. There are millions of other places to get problems if more practice is needed. I'll note that in hindsight, this is one area in which we could have done better with the original AoPS books - there aren't enough easy exercises to carry the student at the top 5% in the nation level to the point where he can do the same problems as the student in the top 0.1%. However, there is still a significant range of difficulty. Part of this failing is simply a result of trying to pack so much in two books. We're able to remedy this in our new books by taking the knowledge we spread over 2 books in the original series and spreading it over many more books in our new series. Therefore, our old AoPS books will serve as the reinforcing mechanism as students complete the new books. (Or as a source of extra challenge for students as they read through the new books.)

Fourth, there should be something in the book independent of the text learning that the reader thumbs through the book looking for. In the original AoPS, these were the BIG PICTURE sections which Sandor wrote. I'm sure at least 25% of our avid readers didn't get more than 50 pages into the books before they flipped through the whole book reading those BIG PICTUREs. These glimpses humanize the knowledge, and tell the reader even more clearly, 'There's tons of cool stuff out there, so keep looking.'

Fifth, the book should strive to do more than dispense knowledge - it should strive to teach the reader how to generate more on his own. So, instead of being a list of theorems and corollaries that follow, the book should inspire the reader to understand how these theorems might be developed, and how and why to apply them to new problems. We can find knowledge at Mathworld or through Google. For mere knowledge, books may well become obsolete in a generation. Teaching how to develop and apply knowledge is much harder - this is what a text should strive to do, both on a micro level (e.g. draw radii to points of tangency to form right angles) and on a macro level (e.g. when stuck on a problem, ask yourself what information you have that you haven't yet done anything constructive with). This, I think, separates our philosophy of education at AoPS from that which we see in practice elsewhere - our hope is to teach our students how to learn and think so well that they don't need teachers anymore.