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It was a century ago this past August that David Hilbert addressed the second International Mathematical Congress in Paris and, in what must have been an extraordinary talk, presented 23 problems which he considered to be the most important challenges that research mathematicians would confront in the 20th century. Sure enough, these 23 problems occupied the minds of the world's top mathematicians throughout the 1900's. You could gain instant respect by just saying that you were "working on Hilbert's nth problem." Some of the 23 problems were solved fairly quickly; some lingered for many decades. I was in high school when a young mathematician named Paul Cohen settled the Hilbert's First Problem — the continuum hypothesis — by proving that it was technically undecidable. This was one of many important events that escaped my notice during my high school years. On the other hand, everyone in this room probably recalls the excitement surrounding Andrew Wiles's proof of Fermat's Last Theorem, the most celebrated example of Hilbert's Tenth Problem, and the conundrum Hilbert actually used as a starting point for his lecture. Some of Hilbert's problems, like the Riemann Hypothesis, survive into the 21st century.

Considering the impact that Hilbert's lecture had in the last century, I was fairly convinced that some mathematician would take up the gauntlet and deliver a millennium version of it to celebrate the arrival of the year 2000. That talk has yet to be delivered, and let me assure you that you are not about to get it here from me. Instead, I would like to fill the void in a comparatively small way by giving a fairly insignificant talk about the challenges that I see looming not for research mathematics, but for the teaching of mathematics, or more precisely the teaching of mathematics in the United States of America. I am surely not the best person to do this, but this comes close to being the ideal audience, and the turn of the millennium is clearly the appropriate time, so at least two-thirds of the occasion is right.

I will frame this talk in a metaphor that I think high school mathematics teachers will find highly appropriate, particularly teachers who have taught courses in algebra. Several years ago, I wrote an article called "Climbing Around on the Tree of Mathematics" that compared the body of mathematical knowledge to a tree. In it, I pointed out that the living part of our subject is evidenced in the beautiful foliage at the ends of the branches of the tree, but we teachers spend most of our time teaching our students about the trunk, which is ancient, uninteresting, and hard. My main point in that article was that technology, specifically graphing calculators, was allowing students to get into the branches directly, without climbing slowly up the trunk. There were, however, some additional explorations of the tree metaphor, one of which was that the Tree has grown to the point that it is now much too big for anyone to know in its entirety. I ended the article with a prediction that the next wave of reform, spurred on by technology, would be directed toward the traditional mathematics curriculum — and it is that second wave of reform that I would like to talk to you about today.

The urgent need for reform in our traditional curriculum notwithstanding, nobody wants to make matters worse. That is why my basic premise, in keeping with the tree metaphor, is that we must spend the beginning of the 21st century getting back to our mathematical roots. The roots of the Tree of Mathematics go deep. Very deep. Algebra, geometry, and trigonometry were already ancient fields of study when the last millennium was celebrated, and even calculus, a relative newcomer in the curriculum, goes back more than 300 years. It is tempting to conclude that these topics are fundamentally what mathematics is all about and that they will still be around when Dick Clark watches the ball fall in Times Square in the year 3000. But can we trust the roots of mathematics? We algebra teachers spend more than half of our classroom time finding roots, but we realize that not all of the roots we find are correct; some of them are what we call extraneous. Extraneous roots, which are not really part of the solution, creep in unexpectedly as a result of how we attempt to solve the problem. So I contend that our first challenge in the new millennium is to get back to our non-extraneous roots, if we can first identify what those are.

Consider that in 1900, the year of Hilbert's famous talk, only about 11% of American children between the ages of 14 and 17 attended secondary schools. But that percentage was rising fast, in response to the changes that were already establishing the United States as a world leader in all of the areas one associates with an educated society: technology, journalism, government, and economics. By 1930, the percentage was up to 51, and today 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 when we talk about the roots of mathematics education, that is where the extraneous ones began to creep in.

First, let me review a little history from the point of view of the educational theorists. (This is hardly my area of expertise, but I am freely robbing these facts from "Curriculum Ferment in the 1890's" by Herbert Kliebard, a chapter in a fascinating book, The Future of Education: Perspectives on National Standards in America, published by the College Board.)

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 kids were out plowing the fields. If you have to watch a horse's butt all day, they figured, you might as well get something accomplished.

The situation was consequently ripe for reform when the humanist movement came along, most dramatically in the person of Harvard President Charles W. 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 they wound up dealing with the underlying problem of all those emerging high schools with their different local curricula. The humanists did believe in mental discipline, but they also felt that a rich, varied, interesting, 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, such as algebra, geometry, and the rest of our familiar high school subjects.

The developmentalists, led by G. Stanley Hall, disagreed with the humanists, contending that learning had to be geared to the child, not to the narrow expectations of the college community. They also became the first reformers to question the idea that such college-preparatory subjects as algebra were really the best preparation for life. The developmentalist movement led to such things as required courses in adolescent psychology for teachers and multiple tracking through differentiated curricula for students.

Meanwhile, a parallel fight was occurring over the design of the curriculum. The traditionalists, whose company included the U.S. Commissioner of Education, espoused a curriculum built around the traditional fields of study, feeling that this was the best way to pass on the lore of our culture to future generations. The Herbartians, including the young John Dewey, opposed putting learning in separate compartments, feeling that subjects should be unified through common themes. That debate, of course, still rages today.

Then, while the curricular battles were being waged in the halls of education, a businessman named Joseph Mayer Rice decided to see what was actually happening in the schools. What he saw, of course, disappointed him, just as visits to American high schools routinely disappoint businessmen today. He did not blame the education theorists, though; he blamed the teachers, administrators, and highly unqualified school boards. This theory was, as you might expect, bitterly contested by the teachers, administrators, and highly unqualified school boards. 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 in as efficient a way as possible. These reformers wanted standards, accountability, elimination of waste — and if they could have imagined it, they would surely have called for high-stakes standardized testing. 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. If left to the laws of nature, they feared, education 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. Therefore, social justice, equal opportunity, healthy living, and coping with civilization were the truly essential subjects that would need to be taught and learned if society were to survive.

The mathematics curriculum we have today in American high schools is, for all its ancient subject matter, an artifact of the 20th century. The roots that support the Tree of Mathematics are essential and deep, but the roots associated with the American high school curriculum are, for the most part, less than a century old, and some of them are only there because of the way that the educational community sought to solve a problem. In other words, some of them are extraneous. Unfortunately, it only takes about a generation of growth before one root starts looking as essential as all the others, and so our true roots become harder and harder to find. Nonetheless, some very dedicated people have been digging around in the soil for several years now in a desperate attempt to separate the essential from the extraneous, and some consensus is actually beginning to emerge.

There appear to be two roots to our tree that are clearly deeper than the others. One is a root that I will call Quantitative Literacy, borrowing the term from Lynn Steen, John Dossey, Robert Orrill, and others. The other is the Mathematics Preparation root, which some might call College Preparatory. Quantitative Literacy, which is not easy to define, consists of the knowledge that everyone ought to have in order to be, well, quantitatively literate. Mathematics Preparation consists of the knowledge that everyone ought to have in order to succeed in higher mathematics and perhaps someday to do research in the field. While there ought to be no reason that mathematics education could not draw strength from both of these roots, they have become sort of a yin and yang for our profession, as we re-stage the philosophical battles of the humanists, the social meliorists, and the social efficiency proponents over such topics as calculators and two-column proofs.

Whatever we define them to be, I believe that both of these roots are essential and ancient. Follow the Quantitative Literacy root all the way back, and you will probably arrive at our primitive ancestors bartering over tools and food. People have always studied mathematics because it is useful, indeed, useful to the point of being necessary. Follow the Mathematics Preparation root all the way back and you will find the eternal quest for mathematical discovery, a quest that has always begun with learners seeking out teachers. What makes these roots essential is that you cannot have mathematics without them. They do for the subject what good roots do, providing the nourishment without which life would be impossible. I intend to take a good look at both of them shortly, but first we must deal with all the other roots that we see around us, some of which might well be extraneous.

Consider first the Mental Development root. As previously noted, the 19th-century model considered the development of a strong mind to be the primary goal of early schooling. It is easy to see how teachers would be drawn to mathematical computation as the ideal exercise for developing brains, but does it really work that way? Do mathematical exercises develop the brain like sit-ups develop the abs, or does rote mathematical knowledge acquired through such exercises simply enable one to do more and better mathematics? Plato was so enamored of mathematics that he made it the basis of his entire cosmology, and yet he wrote in The Republic, "I have hardly ever known a mathematician who was capable of reasoning." The implication is that the learning of mathematics, in and of itself, does not produce great thinkers. Sound familiar? Plato also wrote, "Bodily exercise, when compulsory, does no harm to the body; but knowledge which is acquired under compulsion obtains no hold on the mind." About the education of children he wrote, "Let early education be a sort of amusement; you will then be better able to find out the natural bent." Assuming that Plato was reacting to the educational establishment of his day, it would appear that not much reasoning or amusement was going on in mathematics classes conducted under the Grecian formula.

Undoubtedly, kids in Plato's day and in the millennia prior to Plato were taught mathematics as part of their mental development. But let us remember that it was actually necessary for children to acquire a great deal of rote mathematical knowledge in order to do any mathematics at all. Tedious computation, like it or not, was all part of the mathematical game. A youngster in Plato's time who had mastered long division would have done so like you and I did: by spending many boring hours doing rather elementary, non-reasoning mathematics. Regardless of whether the exercise made the brain any bigger, when confronted with a division problem, the youngster who knew the algorithm would have been able to solve the problem and the youngster who did not would have failed. So — did learning the long division algorithm develop the brain, or did it simply open up new possibilities to demonstrate knowledge?

The answer to that question, which is not obvious, is important for determining the nature of the Mental Development root. We all know that today's kids can do long division (and many other manipulations) quickly and accurately on calculators. By doing so, they can avoid the tedious mental exercise that used to be necessary. Now, that sounds like a great idea if the purpose is merely to find the solutions to long division problems, but it's a terrible idea if that tedious mental exercise is necessary, or even desirable, for developing children's brains. The NCTM position on this issue, based on sound research and echoing the humanists from a century ago, is that the stimuli for mental development need not be restricted to the same rote drills that bored our grandparents. A child who divides on a calculator, they argue, can spend more time thinking about reasonable answers and about the connections between mathematical symbols and real-world models. The emphasis can be placed on reasoning, which would probably have pleased Plato. Moreover, they are much more likely to get the right answer, although it is not clear that anyone cares.

Some who disagree with NCTM argue that only a steady diet of rote learning can build up the mental muscle, and that traditional drills on computational techniques are essential in any such diet. These folks distrust calculators, and they consider the inability to do long division without one to be a sign of mental weakness.

My own theory about the Mental Development root is that it is, indeed, essential and not extraneous. We do need to develop young brains, and rote learning is a good way, perhaps an indispensable way, to do it. However, I feel that it is not only desirable but essential that we abandon yesterday's rote learning and replace it at every level with the rote learning that is actually necessary today. There is no shortage of rote learning to be done; in fact, there is far more of it when you consider how much more one needs to know to become an expert on anything, including mathematics. What we must be prepared to admit is that nobody really needs to divide 563,346 by 2,367 without a calculator. Indeed, we need to identify a lot of other things that nobody needs to know, too. The scary thing is that while we let time slip by without giving the world guidance about what sort of rote learning really is important, the world is sending our children the subtle message that all rote learning is equally valuable, no matter what sort of knowledge you wind up with. That is how you win at Trivial Pursuit, Jeopardy!, and, God help us, Who Wants to be a Millionaire. A recent study of the brains of London cab drivers showed that the brain actually grows in size as a result of rote learning: in their case, the part of the brain responsible for charts and maps. Similar data for rote mathematical learning might be right around the corner. But whether they find our part of the brain or not, we should not abandon the role of mathematics in Mental Development, because it is one of our non-extraneous roots.

As long as the subject has come up, let us look next at the Technology root. There are those in our profession who consider technology to be an essential part of mathematics teaching, and there are those who feel that it is a fancy distraction — possibly worthwhile, but extraneous to the teaching and learning of mathematics. The latter group, none of whom probably made it to this conference, would argue that technology is too recent to be an essential root; after all, didn't we get along fine for thousands of years without it?

Frankly, I'm not so sure that we did. Before the electronic calculator and the computer, there was the slide rule. Before the slide rule, there was the abacus. A long time before that, there were fingers and toes. And really, is finding a cosine on a calculator all that mentally different from finding a cosine in a trig table? Is a log table considered technology if we use it to do long division without really doing long division? What about the compass, the protractor, and the straightedge — are they technology? What about the calendar, the clock, and the astrolabe? The history of mathematics is tied inextricably to a long progression of ingenious devices that have helped the mathematicians of their day to do their thing, and it has always been part of mathematics teaching to show the next generation how they work. In that respect, I contend that it is another essential root. We cannot have mathematics, and hence we cannot have mathematics education, without it.

Admittedly, things have changed a lot since the slide rule. Today's technology is programmable, giving it the capability of becoming a different tool in the hands of different mathematicians. It is no longer merely a question of showing the next generation how a computer works; what we really need to do is to suggest the possibilities. We, as teachers, must all realize by now that we have literally no idea how our students will use our mathematics with their technology ten years from now. (Future of Technology transparency) The advances in computer technology have been breathtaking, even in the field of research mathematics. Such recent mathematical breakthroughs as the proof of the Four-Color Map Theorem and the classification of all finite simple groups would have been impossible without computer assistance. Entire mathematical fields of concentration, like linear programming, operations research, dynamical systems, and fractal geometry grew up dependent on computer technology. We cannot possibly teach all this stuff to the next generation, but the good news is that we don't have to; if we simply let them use today's technology to do the mathematics they are doing today, then they will use tomorrow's technology to do the mathematics they will be doing tomorrow. I suspect that you already believe that; otherwise, what would you be doing at a conference "embracing new directions"?

Before we become convinced that there are no extraneous roots, let us look at one of the most respected roots of all: the Core Curriculum root. We live in a world where it is presumed that every educated young person will know algebra, geometry, and elementary functions. If they expect to go to college, they had better know trigonometry and analytic geometry. If they go to college, they will learn calculus. If they major in mathematics, they might learn some other stuff. They might even learn some other stuff in high school, but it will be in addition to that sacrosanct core curriculum.

I used to assume that the core curriculum had always been there, dating back perhaps to the ancient Babylonians, until I discovered that it had actually evolved from the decisions made by the Committee of Ten at the beginning of the 20th century. The object was to prepare students for college mathematics, which meant calculus. Today, despite a century of the greatest growth mathematics has ever enjoyed, that still means calculus, and so the core curriculum, defying all logic, endures in much the same form as it was in when our grandparents went to school.

That does not imply that people have not tried to change it. New discoveries in mathematics and the perceived needs of the workplace have indeed had their effects, but, incredibly, they have only served to add more material to what was already one of the most content-laden subjects in the high school curriculum. So, our bloated, 1000-page textbooks now have sections or chapters on applied matrix algebra, descriptive statistics, transformational geometry, linear programming, computer programming, game theory, graph theory, Boolean algebra, and optimization — topics that were unheard of forty years ago — while only a few of the topics from forty years ago have been dropped. (Log tables and trig tables come to mind. Oddly, so does solid geometry, which was judged to be irrelevant for calculus. Axiomatic algebra proofs, added in the New Math era, were mercifully subtracted 25 years later.) Moreover, the added emphasis on problem-solving has doubled the number of problems and probably more than doubled the background required for teachers to explain them to their students. The sad effect of all this addition has been chaos, with teachers picking and choosing topics from textbooks that are far too enormous to teach, often omitting (or short-changing) topics from the very core curriculum that the bloated textbooks were designed to preserve.

It was only when I started working on a precalculus textbook myself that I came to understand how powerful are the forces working against the restoration of sanity in the high school mathematics curriculum. I was convinced that we could produce a "lean and lively" precalculus textbook that would prepare students perfectly for calculus in approximately two-thirds of the pages typically being used. My first jolt of reality was when we were given the state adoption criteria for four major states. Leaving out any topic on any of those lists was unacceptable to the publisher, as it would automatically preclude adoption in some major state. Needless to say, everything in the current precalculus textbooks was on the list, since that is where the state adoption committees got their lists.

So then we tried to cut down on the redundant exercises. Nope; exercises have to be paired so that every even-numbered exercise without an answer in the back of the book is preceded by an almost-identical odd-numbered exercise whose answer can be looked up. Never mind the perverse message this gives kids about how to solve problems; that is what the publishers have been told that our teachers want. And I am afraid that many teachers actually do.

You also can't show an example in the book if it does not lead to at least six exercises, and you can't give an exercise in the book if it has not been set up by at least one example. If a reviewer has an alternate way of presenting a topic that he or she thinks would be a nice example, you have to add that example. If a competing new-wave textbook has a wrinkle some reviewer prefers, you have to add some version of that wrinkle. If a competing classical textbook has retro stuff for which some reviewer is nostalgic, you have to include the retro stuff, although perhaps only as a margin note. I am proud to say that our precalculus text eventually came out with 835 pages, including appendices and glossary. This is still obscene, but it's 15 pages shorter than the previous edition and a full 153 pages shorter than our main competitor.

Next is a root I call the Competition root. I don't know exactly when mathematics became competitive, but for some reason it is now the most competitive of the traditional subject areas, assuming that we resist the urge to classify interscholastic sports as a traditional subject area. Not only do schools spend considerable time preparing our best students for formal mathematics competitions, but many schools actually create competitions where competitions should not exist, comparing individual students, classes, schools, school districts, states, and even nations, on the basis of mathematics scores taken from various assessments. It has come to the point that we can no longer measure success on the basis of what our students have learned; we must now factor in how many butts our students have kicked on AP tests, or PSAT's, or NAEP, or state exit tests, or departmental common exams.

The effect of all this competition on the teaching and learning of mathematics should not be underestimated. On the low end of the ability scale, teachers in many states spend most of their time and effort on preparing students for high-stakes minimal competency tests. They do this not so that their students might be minimally competent — an unfortunate goal in the first place when you think about it — but so that their students will rank well when compared to other students. On the high end of the ability scale, all of us teach topics that might well fade from the curriculum were it not for the fact that our best students need to know them to answer the classical math contest questions. Such questions, deliberately designed to test cleverness and quickness, are often so far removed from reality that they are not even useful in our mainstream classroom discussions. I once talked to Gail Burrill after she had observed the finals of a national math competition, and she was unsure what to make of the experience. "Those youngsters were brilliant," she said with her customary enthusiasm, "but where was the mathematics?"

Don't get me wrong, now. It is the nature of mathematicians to love a good puzzle, and I like a great problem as much as the next person. I serve on the American Mathematics Competitions Committee and love working with that group. We all know that good students look forward to the AHSME, the USA Mathematical Talent Search, the annual modeling contest, and so on. They ought to enjoy them, they ought to strive to do well, and they ought to take pride in their scores. However, using the results of such competitions to assess the teaching and learning of mathematics at a school is no more valid than judging the physical fitness of that school by the record of its football team. For that matter, you cannot even judge the physical fitness of the football team on the basis of its record, because success in football does not always correlate well with physical fitness, especially among the 250-pounders. The same sort of thing is true with any kind of mathematics competition, in which success does not always correlate well with the teaching and learning of mathematics.

Moreover, while good mathematics might be done in competitions, it is not the competition that makes the mathematics good. Perhaps by concentrating on the essential roots and giving up some of our favorite contest topics we can improve the teaching and learning of mathematics for all students, and ultimately make the competitions better and more relevant. Meanwhile, I am going to call the Competition root extraneous.

The final root at which we will look is more difficult to capture in a pithy name, but I call it the Cultural root. As mathematics teachers we can get so dazzled by the sheer usefulness of mathematics that we forget that we are exposing our students to one of the greatest accomplishments of the human mind — a subject as rich in history as the pageants of world politics and as creative as poetry or studio art. Every world culture has its mathematical stories to tell, stories as old as the pyramids and as new as unlocking the genetic code. This root is far from extraneous and might be the most basic root of them all. There are good cultural and historical reasons to expose students to certain classical results in mathematics, regardless of whether or not they will actually need them later in their lives.

Take the quadratic formula for example. I have often told of my experience of watching Johnny Carson quote that formula by heart one night in his monologue. This prompted thunderous applause and great laughter from his audience, all of whom recognized it immediately as being funny. He then noted that his teacher back in Nebraska had told him to memorize it because he would need it some day in the future. That drew another big laugh from the audience, probably because they saw the next line coming. He finally remarked that he had waited fifty years before he actually got a chance to use it — and that was to get a laugh in his monologue.

Although we still use that line today about needing the quadratic formula in the future, it is becoming more difficult to say it with a straight face. After all, calculators will now find exact answers, real or complex, to any quadratic equation you give them, and if you really want the solution to the general equation

, the TI-89 will give you that, too, even going so far as to derive it for you step-by-step. It seems logical to conclude that, if finding roots is the name of the game, memorizing the quadratic formula is no longer actually necessary. Does that mean that we can toss the quadratic formula onto the same scrap heap as log tables? I hope not. But let us not justify keeping it in the curriculum because our students will need it later on in their lives; let us keep it there because it is a significant bit of mathematics that can actually be appreciated and understood by high school students. Let us keep it because it is part of our mathematical roots, in this case the Cultural root, and, like Shakespeare, it is worth being studied for its own sake.

That gives us, by my count, five essential roots for school mathematics: Quantitative Literacy, Mathematics Preparation, Mental Development, Technology, and Culture. I have also mentioned two extraneous roots, the Core Curriculum root and the Competition root. There are probably dozens of other extraneous roots that we could talk about (New Math, Basic Skills, locally-generated roots, and so on), but I want to spend the last part of this talk focusing on our essential roots and the Mathematics for the New Millennium, as promised. I have tried to lay the necessary groundwork by dismissing the Core Curriculum as an extraneous root, and if you do not agree with that, well, indulge me a little bit longer because it is too late now to turn back. We cannot take mathematics education back to our true roots if that big, fat, imposing-but-extraneous Core Curriculum root is smack dab in the middle of our way.

So how do we get back to our truly essential roots? First of all by understanding them. Sadly, we have been in the extraneous mode for so long that it is almost impossible for us teachers to tell what is essential anymore. It is difficult enough to open up a textbook and teach a course based on what is inside it; it takes real wisdom and experience to identify what needs to be taught. The mathematics community has plenty of wisdom and experience, but for years it has been committed to other pursuits, like abstract measure spaces, under the comfortable assumption that the Core Curriculum in mathematics would suffice to feed the mathematics pipeline forever. We now know that that assumption was false.

What woke most of us up was a devastating 1983 report from the National Commission on Excellence in Education, entitled A Nation at Risk. As the name of the report suggests, this group was not just concerned about declining SAT scores. Their concern was nothing less than cultural innumeracy, the inability of American citizens to cope with the mathematics that would affect their lives, and ultimately, the life of the American nation. The pessimistic report so shocked the mathematics community that they responded with an unprecedented focus on education, churning out multiple studies, projects, and calls to action, culminating in the NCTM Standards and what in retrospect can be called a classroom technology revolution. Reform was on everyone's agenda by 1990, and I don't know about you folks, but my move away from the blackboard made me realize that I had been teaching for fifteen years dressed in the Emperor's New Clothes. There were probably thousands of teachers who were likewise born again in that first wave of reform, but even in the midst of all that change there was one thing that stayed basically the same: that good old extraneous root, the Core Curriculum.

Oh, sure, some of us started teaching more statistics and probability, more modeling, more data analysis, and more graphical interpretation, but these were added on top of what we were already teaching. The authors of the Standards, bless their brave hearts, made a bold attempt to tell us how we might alter the core curriculum by including these controversial pages in that historic document: Topics to Receive Decreased Attention and Topics to Receive Increased Attention. Needless to say, nothing in the original Standards received more violent negative reaction than those two pages. Indeed, so bitter was the experience for the authors that they deliberately refused to include any comparable section in the new Standards 2000 document. It would appear that nobody, not even the folks with wisdom and experience, can get past that extraneous root.

But appearances can be deceiving. It turns out that, in the back rooms, some people with wisdom and experience have been trying for some time to determine what we ought to be teaching to the students of the 21st century. Their strategy has been to ignore the extraneous roots and concentrate on the essential root that we have lost sight of, namely, Quantitative Literacy.

The College Board, for example, has been studying Quantitative Literacy pretty intently for about ten years. A 1997 publication, Why Numbers Count, edited by Lynn Arthur Steen, summarizes the effort up to that point, and a forthcoming book by John Dossey will carry it even further. The Dossey-led committee that has worked on the project first attempted to define Quantitative Literacy. Their definition focused on four key areas (synthesized from an original five):

They then did a study of all the College Board tests, from SAT's to AP's, to determine how well they were assessing Quantitative Literacy in each of those categories. What they found, thanks to our friend the extraneous root, was a huge overload in the "Patterns, Functions, and Algebra" category, most of it tested at the most abstract level. So they developed some questions designed to assess Quantitative Literacy more appropriately and effectively, and they field-tested them in middle schools. They also field-tested teacher reactions to the exercises and found that, as expected, teachers were less comfortable with the exercise the more it deviated from traditional algebra. The results of that study will be included in the forthcoming Dossey publication. It is hoped that this effort, combined with similar initiatives of the Mathematical Sciences Education Board, the Mathematical Association of America, NCTM, and several other professional organizations, will soon begin pushing the American mathematics curriculum in the direction of more Quantitative Literacy for all. I should warn you, however, that history warns us against holding our breath.

It is tempting to blame the paucity of Quantitative Literacy in the core curriculum on the colleges. It was, after all, in service to the colleges that the college-preparatory curriculum of algebra and geometry topics was instituted in the first place. Indeed, if that were all there were to it, we could dismiss all this college preparatory business as an extraneous root and get back to teaching essential mathematics to everyone. But there is an important reason why that won't work: anything college preparatory also affects the Mathematics Preparation root, and that root is just as essential as Quantitative Literacy. We, as mathematics teachers, feel a gut-level responsibility to produce not just a quantitatively literate citizenry, but the mathematicians and scientists of tomorrow. The call to civic duty notwithstanding, we do not want to sacrifice the future of our profession in order to produce lawyers who can understand Simpson's paradox or shopkeepers who can amortize a loan. We must, if you will, answer the primal urge to propagate our own species —~that is, mathematical people. That means that, no matter what cards NCTM lays on the table, the colleges still hold the ultimate trump: the needs of the research community. Unfortunately for us at this moment in the reform effort, the colleges hold that card so close to their vests that everyone assumes they want us teaching that good old Core Curriculum.

You might be surprised to learn that the Mathematical Association of America established a committee way back in 1953 "to modernize and upgrade" the mathematics curriculum at the college level. This committee, called the Committee on the Undergraduate Program in Mathematics, or CUPM for short, is actually still meeting today. You might also be surprised to learn that CUPM's first proposal in the mid 1950's was to change the traditional first-year course (the standard precalculus mix of trigonometry, college algebra, and analytic geometry) to an integrated introduction to continuous and discrete mathematics. Obviously, this did not happen. In the 1950's it was much too radical an idea for the college departments to accept. A more recent CUPM report included this passage:

I mentioned that this report was more recent, and it was: Would you believe 1965? Again, obviously, nothing much happened. CUPM would issue many more calls for change over the years, as they gathered more and more compelling reasons why a richer mix of mathematics would have to be taught to college students. Still, it was only after the applications of mathematics became diverse enough to force the research into brand new areas that the larger mathematics community began to see that CUPM had a pretty good point. By 1990 many universities had changed their undergraduate major from "Mathematics" to "Mathematical Sciences" and had begun offering a greater smorgasbord of courses to their undergraduates. Meanwhile, CUPM has been preparing yet another major report, and a few independent-thinking departments have actually moved beyond calculus reform to reforming their entire curriculum.

The colleges are at a different place today than they were ten years ago, and the implications for the College Preparation root are profound. In light of this fact, I find it particularly ironic that the only people in the universities who seem to be talking about college preparation are the ones who are resistant to change. They have been heaping the Core Curriculum root with all the fertilizer they can muster —and they can muster a lot of fertilizer — in the apparent hope that we might never look beyond the extraneous and discover what today's College Preparation root really looks like. I myself am a poor judge of what the needs of today's mathematical research community are, as I have been out of that environment for 27 years. But I was recently at a meeting where the editor of the venerable American Mathematical Monthly announced that the hottest topic to write about these days for his journal was statistics! If that is true, I am willing to hypothesize that the Core Curriculum, as most of us understand it, is several beats out of step. At the very least, we need to reconsider the familiar argument that "we have too many important topics to teach in this course to waste time teaching statistics."

We face no shortage of challenges as we enter this new millennium. There are still wars going on in the world, the distribution of wealth is grossly uneven, there are environmental disasters just around the corner, and this country is a decade away from a crippling shortage of teachers. In the large picture, mathematics curriculum reform might seem like a challenge best left to simmer on the back burner. But I submit to this group a few cogent reasons that we need to take this challenge seriously:

Speaking of roots...More than half of high school algebra consists of techniques for simplifying expressions or solving equations. The main purpose for simplifying expressions is to solve equations more easily. There are already calculators that will solve equations for you whether you know algebra or not, and if that doesn't convince you that most of our algebra is unnecessary, just wait until tomorrow's calculators hit the market.

I continue to hope that professional organizations, standardized test and contest developers, state standard committees, and textbook publishers will start making the tough decisions about what we ought to be teaching and then give us the resources we need to teach it well. The current curriculum is too large and incoherent to teach well, and it is too uncritically accepted for us to assume that it is what we ought to be teaching. Until that changes, however, our only hope as teachers seeking a little sanity is to get back to our non-extraneous roots ourselves. I therefore urge you all to work with your departmental colleagues to trim your courses down to a teachable set of important topics. Instead of just teaching something because it's in the book, ask if it can really be included among one or more of our essential roots:

Those are the criteria that I plan to use when I am forced to choose between means and medians or the latus rectum of a conic, Heron's formula or Descartes' Rule of Signs, double-angle identities or symbolic logic. Whichever way I go, my students will miss out on some great mathematics — but at least they will learn something essential.