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It is now five years since the publication of the NCTM Standards, and I doubt that I have to tell a reader of this journal that we now find ourselves in some uncertain and exciting times. While we mull over the implications of mathematics reform for our own classrooms, our colleagues are wrestling with the same issues in coffee room conversations and untold other forums -- wherever teachers these days talk to teachers. It is not just that they need to talk about reform; it is more like they hunger to talk about reform. Attendance at professional meetings has scrolled off the viewing window, to use the current hip jargon, and not because of any great increase in the population of teachers. A recent NCTM regional planned for 2500 attendees and was visited by more than 8000. The January meetings of the AMS and MAA are so crammed with pedagogy sessions that it is hard for a researcher to get an abstract measure space in there edgewise. Obviously, there is something going on here, but what? This is not, after all, the first attempt to transform education. The fact that this current attempt at reform has lasted this long and is still apparently growing is at least a local curiosity in the history of education, and just might be a globally significant event. As a mathematician I submit that it is time for someone to model this growth, to predict its future, and, in the tradition of good prophecy, to explain why the model works. I intend to give this a try, even though I am not much of a prophet. Teachers never are, of course; maybe that's why they call schools "non-prophet organizations."

Let us begin by assuming that education in American high schools is a function with zero derivative most of the time. There are good reasons for this, which I will get to a little later, and it probably should be considered as the First Law of whatever model we might propose. Whether we look at education from the point of view of curriculum, pedagogy, assessment, or even goals, we do not see much change in the viewing window of, say, this century. We can zoom in on a few blips here and there, like this one representing the New Math:

Notice that a logistic model does not violate the First Law, as it will level off again after a period of transition. The problem with the logistic model is that there are very powerful forces working within the American high school infrastructure that contend strongly against it, which is why almost all changes attempted in the past have turned out to be "normal." I hope to explain this in a minute, but for now let us simply note that the history of our educational system suggests this about the future: If the current mathematics reforms do result in lasting change, then it will be because something extraordinary will have intervened to render the system changeable in the first place.

I recently had the opportunity to take a trip to Germany. Now, I had never been to Germany before, but I had no doubt that when I got there, I would find McDonald's restaurants. Never mind that Germany invented the hamburger; it is McDonald's that has sold more than 200 billion of them. Sure enough, I was greeted by the golden arches before I even got out of the Frankfort airport! Clearly, McDonald's has been successful in spreading their definition of a hot meal throughout civilization, but if we attribute that success to mere business savvy, then we miss a good story. It was an anthropologist who first pointed out to me that McDonald's has met with unreasonable success, for reasons that have less to do with food than with human tribal culture [1].

Think about it. Since the earliest days of humankind, eating has been a social occasion. As hunter-gatherers, we have been conditioned through centuries of tribal living to come together to share food with others of our kind. From Christmas dinners to Seder suppers, from parish picnics to business lunches, from banquets with hundreds of convention colleagues to a candlelight dinner with a lover, eating has always been as much a tribal ritual for our species as it is a necessity of life. It is therefore no wonder that we are most comfortable when we are sharing our meals with family, or with close friends, or with people with whom we are can share a pleasant conversation. It is why your mother's meatloaf will always taste best, and why hot dogs always taste better to us at the ballpark. It is why two people in adjacent seats on a 747 can fly for an hour without exchanging a word, then suddenly begin talking when the meal arrives. It is why ethnic restaurants thrive in New York City. We don't just eat; we share our culture. Eating is a tribal thing.

Now consider McDonald's. An objective analysis of their food does not suggest that they achieved such great success by serving delicacies that surpassed everything else available. Nor did they achieve international success by selling regional fare; they sold the same burgers and fries no matter where they set up shop. Their success was not gastronomic, but anthropological: They stepped into a momentary void in the continuum of tribal culture and became, essentially, everyone's mother's meatloaf. How did they manage to accomplish this feat?

What happened was that we stopped eating lunch at home. We all did, and for different reasons: Dad was at work, the kids were at schools far enough from their homes that their school had to provide lunch for them in cafeterias, and Mom was either at work herself or at lunch with her friends -- other members of her tribe. Automobiles made it easy to get around, so where the tribe gathered was not a real restriction. What resulted at first was a restaurant extension of the status quo: More people were eating out, but they were still most comfortable in their local restaurant, dining with their colleagues and with the folks we might call "the usual crowd." Later, as people became more mobile and automobiles became extensions of their lives, the most palpable sensation of lost tribal connection hit them when they had to dine alone in a strange restaurant. It was into this opportunity of a lifetime that McDonald's stepped with their golden arches.

They erected these arches, of course, within sight of as many automobiles as they could, specializing in interstate highways and shopping malls. They inserted themselves into your culture within a few blocks of your home, your school, or your workplace, and then, when you were on the road 1500 miles away from your tribe, a stranger hungry in a strange and forbidding land, your McDonald's friends were there with their golden arches to beckon you home to a meal that you would know, trust, and in a strange and primal way, love. You could walk in and see a smiling face in a familiar setting, and you could speak your tribal jargon of "Big Mac, large fries, and a Diet Coke," and they would understand. You could even dine totally alone, and still feel like a part of your tribe. And if your kids were with you, you could savor the humbling sensation that McDonald's is as much a part of your children's tribal culture as your own dinner table at home.

This, then, is the unreasonable success of McDonald's. They have taken the tribal phenomenon of the meal, which has remained so distinctive in different cultures precisely because it is a social occasion associated so intimately with the tribe, and they have fashioned it unto their own image in a paradoxically universal way. While ethnic and regional cuisines still dominate our tribal gatherings at dinner, the Big Mac and the Egg McMuffin have somehow become food for the world. It's a great story, especially if you bought McDonald's stock in 1960 -- but I must leave that story now. I have not forgotten that I am writing here about education reform.

There are not many human pursuits that are analogous to eating in the life of the tribe, but I would submit that education is one of them. Every human culture feels the obligation of passing on the accumulated wisdom of the tribe to succeeding generations -- that which we would call the "lore" or "tradition." In simpler times, the process of doing that probably sufficed as a definition of education. Parents taught their children most of what they needed to know; then wise elders and sages passed on further learning to larger groups, much as teachers do today.

The early history of education in the United States began with this simple tribal model. Education existed to pass on the accumulated lore, and much of it occurred within the family. Schools and teachers were community-based, still very much in tribal units. High school students, of course, were not in school at all; they were out in the fields or otherwise engaged in what we now know as "work," something which today begins at a much later age. That all began to change in the early 1900's with what seems in retrospect to have been a remarkable proliferation of secondary schools, bringing the percentage of 14-to-17-year-olds in schools from about 7% in 1890 to more than 50% in 1930. That percentage continued to grow, and now we have what amounts to universal public education from grades 1 through 12, at least in spirit. In 1990 there were over 110,000 schools in the business of delivering that education, under the watchful eyes of roughly 15,000 separate school districts. You might think that the fact that all of this educational infrastructure has grown up in less than a century means that not much planning could have gone into it, and you are right, but that is not because nobody has been paying attention. Well-established groups of educational theorists and curricular scholars began arguing decades ago about the direction that education should take, and those very arguments continue to this day. Most of them would sound very familiar to current reform enthusiasts. The curious thing, however, is that while their debates raged in the pages of journals and in the halls of universities, the schools themselves were struggling heroically, against all odds and in the face of all logic, to retain as much as they could of that simple tribal model.

The prevailing theory of education in the 19th century was that of mental discipline, which held that the mind was like a muscle, to be strengthened in various ways by various learning exercises. As with exercising the biceps, the learning activity itself was just a means to building a stronger mind; hence it did not matter if the activity was dull, repetitive, or not immediately useful. No pain, no gain. Needless to say, this did not contribute to the popularity of schools, which may have been why more than 90% of the high-school age youth were happy to return to the plows.

The situation was consequently ripe for reform when the humanist movement came along, most dramatically in the person of Harvard President Charles Eliot, chairman of the notorious "Committee of Ten." This group was formed by the National Education Association to deal with disparities in college entrance requirements, but wound up dealing with the underlying problem of all those emerging high schools with their different tribal curricula. The humanists did believe in mental discipline, but they felt that an interesting, varied, and useful curriculum would be just as effective in building the mental muscles as rote drills. They also believed that the ideal college preparatory curriculum (which they had been charged to set up) was also the best preparation for life, and that all students should therefore study the same subjects.

Then along came the developmentalists, who begged to differ with that particular premise. Their philosophy was that learning had to be geared to the child rather than to the narrow expectations of the college community. They also questioned the idea that such college-preparatory subjects as algebra were really the best preparation for life, feeling that it was more effective to place students in tracks that would match their own needs and abilities.

Meanwhile, a parallel fight was occurring over the design of the curriculum. Many educators, notably the Commissioner of Education, favored the traditional fields of study as the best way to pass on the lore of our culture to future generations. This premise was opposed by the Herbartians, who felt that learning should never be carved into separate subjects, but rather unified through common themes.

Then a man Joseph Mayer Rice decided to see what was actually happening in the schools. What he saw, of course, disappointed him; yet he did not blame the education theorists. To him, their debates were basically irrelevant -- to what he saw as the real problems, which were the teachers, administrators, and highly unqualified school boards. This theory was, as you might suspect, bitterly contested by the teachers, administrators, and highly unqualified school boards, and the battle lines were drawn once again. From Rice's studies there emerged another reform movement, the social efficiency movement, which sought to improve the schools along the model of a business producing a quality product. These reformers wanted standards, accountability, elimination of waste -- I think you know the tune. Since there were things in the social efficiency agenda that could offend both humanists and developmentalists alike, the fighting was really getting interesting now.

Finally, into the fray stepped the social meliorists who argued that the real purpose of education was to create a better and more just society. They feared that education, if left to the laws of nature, could go the way of Social Darwinism and create a widening gulf between an educated elite and the uneducated masses. This, in turn, could lead to the disintegration of democracy and of our American way of life. Placed in that context, such questions as "who should study Shakespeare and when should they do it?" took on a new INsignificance. First and foremost, argued the meliorists, schools and curricula would need to be structured to save future generations from the perils of ignorance. Racial equality, social justice, equal opportunity, and coping with civilization were vital subjects that would need to be taught and learned if society were to survive.

I doubt that anyone could have read that brief history of our turn-of-the-century educational skirmishes without hearing strong echoes of the battles that are occurring right now. My school, for example, forms a curriculum committee every eight years or so, so that some unfortunate group of my colleagues can spend six months wrestling with each other over core requirements versus electives, tracking versus no tracking, required summer reading versus free choice, the daily schedule, the yearly schedule, sports and extracurriculars, and myriad other issues. What invariably emerges at the end is some sort of compromise among the humanists, developmentalists, social efficiency proponents, and social meliorists, and the compromise invariably looks a whole lot like the status quo. I myself am a compromise, having found myself arguing from all four of these viewpoints at one time or another this semester. This never-ending debate is one good reason why education has a zero derivative most of the time. Now, however, I would like to look at what I believe is a more important reason for that zero derivative, and for that we must leave the philosophical battleground of the educational theorists and return once more to the schools themselves.

You see, no matter what the educational philosophers were fighting about outside the high school walls, the schools themselves were doing what came naturally to them, and that was carrying on with the primal directive: Teach the children the lore of the tribe. As schools became larger and more centralized, "neighborhood" schools became "district" schools, making the tribal connection less apparent and the education process less comfortable for the participants. The schools therefore adapted, perhaps unwittingly, evolving ways to simulate tribal attachments of their own. Schools fomented "school spirit" to show their tribal unity. There were tribal colors, tribal sweatshirts, tribal mascots, and frequent competitions with other school tribes. Many early schools had tribal uniforms, and some still do. Parent involvement was welcomed from the elders of the tribe, and whether or not it was a community school, people spoke of their "school community." To capitalize on that other tribal imperative, there was naturally a place in the school for the tribe to gather and eat. But most importantly, when it came down to the actual teaching, it still came down to a sage addressing a small community of learners: a teacher talking to students in a classroom.

The schools had to adapt in another way, too. While the learning communities had gotten bigger, the lore of the tribe had simultaneously become a good deal more complicated, so that it was soon impossible for a single sage to pass along everything a young tribesperson should know. It was therefore natural to get your math from a math teacher, your history from a history teacher, and so on. This separation of subjects eventually became institutionalized with the creation of the Carnegie Unit, but I contend that the reason it has resisted reform from the Herbartians and their ideological descendants for nearly 100 years is that this is the path that the tribal model dictated. The effect, in any case, was to drive the sages further and further apart -- even within the same school -- and thereby make institutional reform all the less likely.

Attempts at reforming American education as we know it have come and gone for years with little impact. Originating from outside the tribal model, they have either withered against the will of the tribe or perished in the quest for consensus. Integration by enforced busing is an interesting case in point. A reform from the social meliorist camp, this noble idea was rejected by blacks and whites alike, who pleaded for better "neighborhood schools" instead. New Math, brought in by the humanists against the howling protests of the developmentalists, was perceived by the sages in the classroom as being outside the lore of the tribe, and was consequently doomed from the start. Sex Education, still being redefined every few months by the social meliorists whose agenda it is, endures constant criticism from tribes who fear that their family values are being distorted by the education process. And so it goes.

Needless to say, there is nothing that flies in the face of the human tribal instinct like national directives on how things ought to be. And if those national directives impinge on the critical tribal social rituals, like eating and education, then you had better clear the streets. Consider the sad story of the American History Standards. A group of well-intentioned history teachers met for several years in order to produce the National Standards for the study of American History. During their deliberations, the give-and-take between the humanists and the social meliorists was predictably spicy. In the end, the amount of social melioration in the proposed course was enough to infuriate several conservative politicians, so that before they even got to the schools, the American History Standards were being trashed on talk radio all over the country. The English standards, which have just been released, are only beginning to attract the attention of millions of tribal critics, all of whom feel eminently qualified to pass judgment since English is their tribal language. We have not even begun to see how the tribes will react to a national testing program to see who meets the standards and who does not. Still, this must appear to the politicians to be a piece of cake when contrasted to balancing the federal budget, so they press bravely on.

If you believe anything that I have written so far, then you ought to conclude that, all things considered, it is amazing that the current mathematics reforms are working at all. While the History Standards lie wounded in the water, the NCTM Mathematics Standards are sailing merrily along with barely a scratch in their hull. They have already outreformed the New Math and the Back to Basics movement, and the only tribe that stand actively in resistance to them seem to be the Saxons. So how is this happening? How did 100,000 autonomously-functioning tribal schools suddenly buy into a national movement? Why is this reform succeeding after so many others have failed? I submit that it is because the NCTM Standards, just like McDonald's, came onto the scene at a singularly opportune moment in the evolution of our tribal culture -- in this case with respect to education -- and that they could eventually redefine education just as McDonald's has redefined lunch.

The first reason for the success of the Standards is that John Dossey and the NCTM leaders did a very wise thing as the document was being developed. They took advantage of the large national membership of NCTM, a happy side effect of the large proportion of the traditional curriculum devoted to mathematics, and they solicited input and feedback from teachers every step of the way. Unlike the New Math, which came from ivied halls outside the tribe, the NCTM Standards came through the tribal sages and became established as part of the lore. Moreover, in the case of mathematics, the non-sages of the various tribes had actually lost track of what the lore was. They could still get worked up about Sex Education, but when it came to algebra and geometry, most people had to defer to the local sage out of ignorance.

Still, ignorance alone has never been a sufficient deterrent to keep some members of the tribe from criticizing change on general principles. Recall, for example, the recent involvement of the politicians. This final anomaly, the lack of effective resistance from the fringes, is almost beyond understanding -- until you realize that the explanation is technology, which is beyond understanding as well. The computer revolution has been so sudden and so pervasive that even the most stalwart traditionalists in the tribe believe that a significant rift has occurred in the smooth continuity of the lore. Here, quite obviously, is an entire body of brand new knowledge that the elders of the tribe recognize as important, and they realize that they must trust the sages to pass it on. They have welcomed technology into their midst, and like a Trojan horse, it has brought with it a host of other reforms, from curriculum changes to cooperative learning. It is difficult to imagine how mathematics education in American high schools can ever be the same again.

To me, the interesting thing to watch now is the effect that these other reforms will have on education in general. Circumstances may have allowed mathematics to get their Trojan horse in the door first, but now that the reforms are there, they are becoming part of the lore: There are graphing calculators in chemistry classes, cooperative learning in history classes, and new forms of assessment across the board. Can the reforms that I claim are succeeding in mathematics for reasons that are peculiar to mathematics, also succeed throughout the curriculum? Well, I noted before that the elders of the tribe had lost track of the lore in mathematics. Perhaps the sad truth is that the majority of them have lost track of everything that is being taught to their children by the sages of the day. If that is true, then we could really see some rampant reform in the next few years. I offer you two tantalizing observations: (1) it has been conservatively estimated that the amount of available information, one terrifying measure of the potential lore to be learned, is doubling every ten years -- so that everybody has surely lost track of something; (2) there are many more tribespeople who associate their high schools with sports teams than with any academic department. Perhaps the attention of the tribe has been distracted away from the curriculum, much as our attention to lunch had been distracted just before McDonald's came in. If that proves to be the case, then the sages will have a wide-open arena in which to reform -- if they want to.

It is no wonder to me that the first genuine opportunity to reform education from within the tribal model is coming from the minds of the mathematicians. They are, after all, the sages who are entrusted with dispensing the lore of mathematics -- the subject that is paradoxically both the oldest and the newest in that classical repertoire that the humanists called the "windows of the soul." Mathematics is ancient enough to be chiseled into the Rosetta stone; it is universal enough to carry our message to unknown civilizations on the Voyager spacecraft; yet it can redefine itself in a single generation when one man stands on the shoulders of giants. The beauty, the utility, and the singular importance of mathematics have been unquestioned since the dawn of human cognition, and will continue to be unquestioned as long as tribes send their children to the feet of their sages. Whatever difficulties we may encounter in trying to define what it is that we should teach, we know that there will always be a world hungry to learn it.

Long before McDonald's ever dreamed of a golden arch, there was a mathematician pondering its equation. Long before people flocked to buy Big Macs, they flocked to mathematicians for explanations to the mysteries of the cosmos. And long before McDonald's ever started on the mathematical journey that is commemorated on their famous sign, it could truly be said of mathematics and humankind that there have been "billions and billions sold."