What you had with the original volumes was a great format, and the newer books are almost perfect with the addition of the introductory problems. In my opinion, even adding the little "extras" was a bad decision.

Sadly, I don't think your books will be used much in standard public school classrooms. I have no doubt that they cover the requisite material, but the teachers cannot teach at that level and students do not want to learn at that level.

Graphing calculators and trivial problems are just rampant in my current book. The calculator is treated as the magic device. For example, I know how to take determinants by hand (or knew how to ), but for other people the matrix chapters were just "How to use a calculator 101".

Useless fluff is also rampant. In an effort to see how math can be applied to physics, say, they make us do completely meaningless things like calculate averages of tables of data, instead of meaningful things like using a differential equation to model kicking a guy out of a plane and factoring in air resistance...

In general, my opinion is that math teachers shouldn't even pretend what students are learning will help them in their daily lives, beyond basic stuff like adding or finding which food product is cheaper. Math, like Latin, is great to learn not because of its practical applications, but for teaching how to think and how to learn.

But anyway, the problems aren't so much with the textbook manufacturers as they are with the public school system and surrounding bureaucracy. They wouldn't sell anything if there wasn't massive demand for the junk they produce. If it wasn't for my family's lack of funds ( babies, especially my new little brother, are financial suicide ) and advanced scholar program that will allow me to take free classes at the local (top 15 or whatever in the nation) college, I'd be filling out applications to Exeter and Andover like a crazy man.

Yeah, this is a good point - the big publishers are responding to the market, not setting the curriculum. Frankly, it's probably the NCTM itself that's a big part of the problem, but I don't know exactly how various policy decisions actually get made regarding curriculum. Certainly, though, the NCTM has set a very clear tone in the embracing of calculators.

As for the totally useless fluff in books, I don't think that comes from the NCTM. I'm not sure what drives that.

They just randomly had some info on Nolan Ryan????????

I mean, he's cool, but they must have had some other stuff to randomly put him in there.

jorian

I really like those, they often contain very interesting things.

But not math related.

Do I sense some advertisement?

But yeah, like what else can you say. The math curriculum is so flawed sometimes I wonder why they even bother to teach it. Middle and High school teachers don't know what they're doing, most of the time. I've come across TWO middle school math teachers, who really know what they're doing when it comes to proofs and creative thinking/problem solving. One is Tatyana Finkelstein, who teaches in my district but not my school, and the other is the all-powerful Ravi B. Mr. Frost emphasizes problem solving in his high-level class, which I think is interesting, but I think his main weakness is that he alot of the time tells us things without telling us why all the time, a "they look collinear" type of argument (that's exaggerated).

My 8th grade teacher (who I thankfully don't have to listen to), is clueless. Absolutely clueless. Like it took her a whole day to convince her that you have to switch the inequality sign when you divide by -1. Also, in an extra credit sudoku that I did, there were some expressions in the boxes instead of numbers, and one of them was a really routine integral, she didn't know what the heck it was. According to Mr. Frost, in order to get a degree in Math Education, one has to take Calculus I-IV. I somehow don't believe that they didn't teach basic definite integrals in Calc II, at the very very least.

So yeah, you come to the rescue. For absolutely cool people like us.

We need people like you to motivate a change in a decadent system.

I was reading Feynman's "Surely you must be joking..."

and he wrote an entire essay on how the educational system in Brazil, when he lectured, was completely corrupt. Students memorized, but didn't understand.

I see this reflected so much in school today. People in any math class don't care to understand how formulas are derived and they care less about playing around with the new concepts they learn. All they care about is passing the next test, and that involves inputting all the known formulas in mankind into their TI-83s and during test time, figuring out which ones to apply.

It doesn't help that it's these same students who actually help some of the teachers in our school teach. Because then it's just a cycle of obliviousness. And like all things, shrugging off deep thinking impacts creativity and problem solving abilities in the long run. It's malignant.

However, I'm a believer that human self interest and politics drive education (not something more purposeful), so that also makes me a cynic, in that I believe it's not possible to reform the rotting system we have on our shoulders today... but hopefully I'm wrong about this.

Have you read Liping Ma's work comparing American and Chinese math teachers?

There is a review here:

http://www.ams.org/notices/199908/rev-howe.pdf

One of the most striking results is that Chinese teachers of elementary mathematics have a much deeper and richer understanding of what they teach, despite the fact that they have fewer years of college level preparation, on average, than American teachers. (If I recall correctly, elementary school teachers in China only need two years of college. An American school teacher typically needs four years at least, plus a master's within the first five years of teaching, in some states.)

Of course, there's a big chicken and egg phenomenon in this country. College professors are constantly complaining about the poor math preparation of their students, but those same professors are often the ones with responsibility for having trained the high school teachers who taught their students!

If you get primary and secondary math education "right," you may not need a lot of additional formal postsecondary education to create great teachers.

And if you don't get it "right," it may be very hard to remediate the damage in college programs for training those who have had poor K-12 math education to become good teachers of the next cohort.

There are, of course, always wonderful and heartening exceptions. I have had the pleasure to know and work with many wonderful American math teachers.

The NCTM Standards are not so much to blame as their implementation, in my opinion.

I am very impressed with Marilyn Burns' book, About Teaching Mathematics K-8, as well as the books she wrote directly for children, Math for Smarty Pants and the I hate mathematics book" Terrible titles, but great content in the books! I also like Jan Mokros book for parents, "Beyond Facts and Flashcards" about ways to integrate math education into every day family life. I think all those books are very consistent with NCTM Standards and they have been a great resource to me and to many thoughtful teachers, parents, and students. (Interestingly, all of the above are relatively inexpensive black and white paperbacks, though the children's book do have engaging illustrations done as pen-and-ink line drawings. No snazzy color photos of irrelevant big name baseball players!)

The books I listed above do NOT have flasy stickers slapped on their promotional material saying "NCTM Standards" on them, but they are clearly philosophically compatible with the NCTM standards.

On the other hand, I have been distinctly unimpressed with the standard K-12 textbooks which splashily promote themselves as "conforms to NCTM standards."

Indeed, when I was on a school district committee charged with selecting new elementary school math textbooks many years ago, I wrote to a long-time public schoolteacher I greatly respect and admire (who is a big supporter of NCTM Standards) and asked her what she recommended we look at--and she wrote back, in a disheartened way, that none of the available textbooks really conformed to the spirit of the NCTM Standards, splashy stickers notwithstanding.

The key to great math education, in my opinion, is teachers and/or books and/or peers and/or problems that inspire students to take ownership of their mathematical education.

Standard K-12 textbooks don't do that. I've seen AoPS books do that for some students. I've seen Martin Gardner's books do that for some students. I've seen Marilyn Burns' books do that for some students. I've seen the Number Devil do that for some students.

Sometimes, what I really think a lot of kids need is not a math teacher but a really thoughtful and sensitive librarian with a good knowledge of what's out there and what might work for a given student or small group of students (as well as a collaborative culture of inspirational slightly older or at least more experienced role models willing to help out.)

By the way, Richard Feynmann wrote a sad but funny account of his time serving on a California statewide textbook selection committee. If you haven't read it, I highly recommend doing so. It's available here:

http://www.textbookleague.org/103feyn.htm

Thanks for all the great info (the Feynman story is a classic). Question: Where in the implementation of the NCTM's standards is the problem? While the standards themselves are probably not the problem, I think the culture is - it seems like the culture shifts from buzzword to buzzword, trend to trend, and there's less math with every step.

I'm not sure that the problem lies in either the textbooks or the NCTM. It has more to do with general education in the states. Before we can address the attitude that high school students have towards math, we should look at their apathy towards education. I can't think of a single academic discipline (Math, History, Science...) that has achieved widespread acceptance among its students.

I suppose the best way to find out what causes student apathy is to ask those precious few good middle and high school teachers why they believe that their students don't care. If teacher A believes that it's top-down school policy and standardized tests that are ruining the system, then we need to get people into positions to change that (by "we," I mean like-minded people, not AoPS in particular). If teacher A believes that it's failure on teachers B, C, and D's part to keep up student enthusiasm after they leave teacher A's class, then we need to try to get better teachers at that level. If the students just come into class apathetic already, we'll need to ask questions at the elementary school level. More likely than anything else, it's a combination of the above three. In any case, I can't accept that the general stigma against math is a result of the NCTM or bad textbooks alone.

As a side note, with regards to math education in China, as far as I understand it, the Chinese put a lot more value in education than the states do (please correct me if I'm wrong about this). However, education also serves a different purpose in China than it does in the states. Due to all of the power shifts and political turmoil in China's history, the Chinese people haven't been able to rely upon the government (whether that government be Local lords, the Emperor, or the CCP) for social security. Instead, a Chinese person's "social safety net" is that person's children. When one is old and unable to continue working, they rely upon filial piety and success to sustain them. Because of this, a child's education is seen as extremely important in the Chinese eyes, but the actual educational benefit is secondary to the safety earned as a result of having a child who will be better suited for the workplace than his peers. With this in mind, the Chinese would naturally hire better teachers and make education a more enticing profession. However, whether their educational system is out and out better is an open question. I should mention that, since I'm posting this between classes, I have yet to read the study comparing the US and Chinese math educational systems linked by sophia.

Indeed, I think you are right--it is the culture.

The AoPS community is a remarkable subculture, in which students have created a thriving mathematical community that inspires and challenges many participants. More generally, math teams and math circles are subcultures.

Many families are their own little mini mathematical cultures, whether or not they are actually officially homeschoolers. Sarah Flannery and her father wrote about one such family in their remarkable book "In Code." One thing that struck me was Sarah's story of growing up watching her father argue with another mathematician as they worked out research problems on the kitchen whiteboard. It was eye-opening and encouraging for her to learn that even seemingly omniscient grownups made mistakes on the path to solving problems.

The question is: how do we broaden the reach of those cultures to reach a much greater number?

The NCTM has a slogan: mathematics is everbody's second language. A language is very much a cultural thing--it dies out if you don't have others with whom to speak it.

There are children who grow up hearing that second language of mathematics spoken in their homes, like Sarah Flannery. There are children who may not hear that language spoken at home, but who have the good fortune to become engaged in a vibrant thriving math community (like a mathcounts team or AoPS or a summer program that shows them a different side of math.)

Then too there are remarkable autodidacts like Richard Feynman, whose parents weren't particularly mathematical (but who encouraged his lively curiosity and were great storytellers!) and who had access to a good public library with books he could use to teach himself math. (Though he too credits his experiences on a city math team and in his MIT fraternity later on.)

So, how to change the culture? Outreach. I think AoPS is a force in a good direction.

Culture is definitely the key. I think this is part of what has me somewhat depressed coming out of the NCTM Convention. There was little in that culture focused on math. We'll be trying a homeschool convention and a charter school convention in the next few months. Maybe they'll be more focused on the content.

I am not sure the public schools are capable of being reformed, history both past and present speak against it. I think the trend of more private schools (of all kinds) and certainly the very rapid rise in homeschooling seem to suggest people are leaving the public schools in ever increasing numbers. Richard, you have highlighted the very poor quality of math textbooks, however the problems don't stop with math instruction. I have some very popular 19th century readers (McGuffy's Readers) and most college students cannot read or write at the level of those readers (they were geared for what would be elementary and middle school aged children). The current ability of our children to write is probably the worst of all three subjects. Given this ever downward trend and the massive bureaucracy associated with most schools, I don't ever see significant change occuring.
Just a note on one of your comments about potentially receptive audiences for the AoPS material, yes homeschoolers could very well turn out to be a windfall for AoPS. Understand that many homeschoolers do tend to talk to "those who have gone before" with the result that most homeschoolers have heard of and use Saxon Math, MathUSee, and Singapore Math (which is a great program). However, word gets around fast among homeschool families, so once they learn about the superior materials from AoPS my bet is they will start to migrate in droves (the kids capable of handling it, not everyone!!) We shall see at this years NCHE Conference!!

On the topic of the market for the AoPS books:

I'd agree with Eric on the homeschool market. Those presently using Singapore Math (1-6) could easily move into your Algebra and other texts after completing Singapore 6B. There's only one or two ways to get the Singapore Math books so it might not be difficult to directly market your books to those customers. Those presently completing Singapore 6B can move to their "New Elementary" math series to learn algebra - but these books are not as good as the AoPS books.

I'd also agree that you'll never get a significant number of schools to "adopt" your books for teaching their math curricula.

However, I wouldn't give up on school sales. You can sell a very large number of books to schools with extra-curricular math programs/teams/etc. Parents of kids on the team will often buy the books so the kid can practice at home.

Teachers/coaches sometimes have budgets to buy such materials for team members. Schools on the AMC Honor/Merit Rolls for high AMC 12 or AMC 8 scores might make good places for your marketing efforts.

You need to demonstrate quickly how your books can usefully be integrated into math team activities, sales. Any step-by-step materials that make it easy for a marginally-qualified, marginally-motivated teacher to sponsor a math club will help. Make it clear that top students can self-study from the books with minimal help from the teacher. For students who need help, teachers with minimal math skills will appreciate the solutions manuals. The biggest fear of teachers who might support math clubs is that they won't be able to solve the problems.

Overall, I'd say the extracurricular school market for the books might be at least as large as the homeschool market.

Other possible markets (not sure how realistic, but some ideas off the top of my head:)

Other countries. Canada and Asian countries might be a good start. Your books have a possible advantage of being written in English and yet offering substantial problems. Those seeking to improve their English skills in a problem-solving environment might consider your texts. Cost might be an issue in some countries.

Usually they are math related or have math history.

FOr example, in the circle chapter, one of them was

I strongly agree with Eric. The AOPS books will spread like wildfire among homeschoolers within the next 5 years.

Which HSing conferences are you planning on selling at? It might be useful to have some HSing volunteers from the AoPS booth to help out. We are a weird and quirky lot, but we know how to talk to each other.

We sent for some information about a conference in Sacramento in mid-August. I think the organization is the Homeschool Association of California. We haven't heard back from them yet. We'll check out the Link one next year, too.

Might some of the "algorithmic" method of problem solving stem from the NCLB requirements? If every student has to perform 'above average' (*cough*) to meet NCLB, teaching an algorithm rather than teaching how to actually solve a problem is probably the quickest, easiest way to reach the most students and get them passing. Also the NCLB standardized tests that the students must pass typically won't have the "hard problems" the AoPS community revels in, so it becomes another point of teaching to the test.

Sadly, when an effort is made to bring everyone to average, that includes bringing the top down as well as the bottom up. Thank goodness for the 'extras' that some teachers and AoPS provide the top. It's too bad the 'extras' are just that and not the standard.

NCLB is definitely exacerbating the problem, and swiftly pushing things in the wrong direction regarding teaching problem solving versus teaching 'memorize the algorithm'. Though exactly why this is the case is very much open to debate. For example, if the tests that were used to evaluate schools were structured appropriately, they could encourage teaching how to creatively solve problems as opposed to memorizing algorithms. And, if schools weren't evaluated on a single number of students who clear a minimal hurdle, but instead on each student's growth from year to year (where a brilliant kid getting much more brilliant counts for at least something), then NCLB might be a force for encouraging the inclusion of more challenging materials in the curriculum. Alas, neither of these are the case, and what we're left with makes NCLB part of the problem, for much of the reasons you mention.

A lot of states, including NC, say that they are measuring "growth" and rewarding teachers and schools that meet "expected growth" targets. The problem is in the measurement. They give all students the same relatively easy test over grade level curricula then claim that they can scale scores in such a way that "growth" is measured. Unfortunately, it's not quite that simple.

Teachers are rated by the "average growth" shown by their students. Those not meeting "expected growth" are spanked with additional training and student testing requirements.

If good ways to measure growth could be established, then the proper incentives might exist for teachers to challenge all students at an appropriate level. Those writing the standardized tests, developing growth measures, and analyzing the data have the power to create the incentives for challenging students that we would like to see.

I suspect that the real reasons that we don't see real growth measures are philosophical and political.

A.Concerning the difference between your textbooks and other textbooks:

1)Hard problems
Well, if the other books don't teach you to do the hard problems nonformulaically, it is not unexpected of them to not give hard problems.

2)Not teaching problem solving.
Very legitimate, and unfortunate.

3)More prose in your books
I'm only closely familiar with the volume one, so you may have addressed this some in your introduction books; I personally felt like volume one had too much prose for most students I worked with.
This is why I used your books primarily as a reference, and not as something to directly teach from. Maybe a quarter of them could actually read through volume one on their own.
My students are young (11-12) and not necessarily very gifted, so this is probably less of an issue for older or smarter kids.

That being said, for Art of Problem Solving to become a viable large-scale textbook, I think it could use to be diluted. Right now its primary purpose is to teach the kids that already seek it out, and it serves that well. To apply it less voluntarily to a larger group, I think it needs to appeal more to that group. To be honest, the appeal would be just as much to the teachers -- an amazing book over a teacher's head is rarely useful in a classroom (There's an analogy of a biology teacher trying to teach physics from a book, something that happens in public schools unfortunately often).

4)More fluff in other books
Some fluff can be useful -- I could never get through a philosophy textbook, but I found Zen and Motorcycle Maintenance and Sophie's World to both be very accessible (both alternate as a philosophy textbook and a novel-like story). One of the best uses of fluff in a mass textbook that I've seen was in Harold Jacobs' geometry book.
Nolan Ryan grabs a kid's attention -- but distracts from the math, which is what you hope to eventually grab his attention.

B. NCLB

I read in Newsweek the other week that Finland has something not entirely dissimilar from NCLB (they are proud not only of their amazing average education, but of the one of the smallest education gaps between top and bottom of the class). The difference there is that the level of "not being left behind" is substantially higher. While it's a wonderful cause to not limit the possibilities for gifted kids (one single copy of AoPS in a math classroom for the exceptional kid to be able to work on his own is a start), raising the level of material for everyone and not worrying about whoever falls behind is obviously not an answer either. Some of the high school classes that I've seen teach algorithms that are already too hard for the students to do anything but memorize. If they are tested to these standards, the high school teacher is left with little choice but go through the algorithms.

I actually think the standards could benefit to be lower in terms of the amount of material, and higher in terms of the amount of thinking. This is politically unacceptable, because the "stupider" kids can be taught the material at their level, but they're at a "natural" disadvantage at thinking. Largely, I think, because they are not taught to think.

Perhaps another way to fix this is to introduce basic physics problem solving into lower grades, and to make learning physics more widespread (that'd be my personal vendetta). Some easy logic puzzles and conceptual reasoning in elementary school may also help. As is, addition, subtraction and especially multiplication tables are a foundation stone of "memorize, don't think." And some of it may just be the genetic factor that we will never be able to help.