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When I was in graduate school umpteen years ago, I was occasionally visited by anxiety attacks centered around such thoughts as "What the hell am I doing here?" Specifically, I wondered how someone as obviously inconsequential as myself could hope to contribute any original thought to the vast lexicon of original thoughts known collectively as Mathematics. Without that obvious prerequisite, what would I use as a dissertation? One afternoon, while I was suffering such an attack in the office of my thesis advisor, he consoled me by suggesting that the entire body of Mathematical Knowledge was very much like a tree. There was this big trunk of general knowledge, from which protruded various branches of concentration, from which emerged smaller branches of specialization, from which finally sprouted various twigs of truly arcane trivia. All that you had to do to expand the tree was to ascend the trunk, climb out on a branch, crawl along some branchlets to reach some twig, then reach out and extend that one little twig by some tiny amount. Doctoral dissertations, in other words, were not about branches; they were about twigs.

Encouraged by this clarification of my mission, I returned to my studies with renewed optimism. Eventually I climbed the trunk to the point where I could access the branch of Combinatorics. From there I shinnied out to the smaller branch of Combinatorial Geometries, found a twig called Factorizations of Combinatorial Geometries, and tentatively squeezed forth a twiglet called Majors of Factorizations of Combinatorial Geometries. That twiglet might eventually bear some kind of fruit, but I won't be there to see it; I long ago retreated back to the safety of the trunk, and here I am -- a high school teacher.

You have to admit that this is a remarkably accurate portrayal of how the body of Mathematics grows. Still, we have hardly begun to explore the richness of the tree metaphor if we limit ourselves to growth. In fact, this is a remarkably accurate portrayal of how the body of Mathematics works. The researchers who are recognized as doing the serious and important mathematics are laboring at the ends of the branches, while those of us who aspire to teach novices are coaxing our students up the trunk, praying that someday a few of them might be inspired to climb past us on their way to exploring the richness of the foliage beyond. The fact that the trunk has not changed perceptibly in centuries of growth does not concern us, nor do the trunk's unfortunate characteristics of being hard, rigid, unyielding, monotonous, and increasingly far removed from the beauty at the end of the branches. Why should we mathematicians, generally respected for our intelligence and perception, fail to be concerned about these things? It's because we realize that there is no access to the branches except through the trunk -- for that is the foundation of the tree -- and the safest path up that trunk is the same path along which we ourselves climbed decades ago.

If that makes sense to you, and it certainly ought to if you have devoted your life to teaching algebra, then let me remind you that it makes no sense at all to the millions of educated people who have decided, most of them since high school, that they have no use for mathematics. They tried to climb our tree, but they just couldn't get their hands around that enormous, intimidating trunk. Don't worry about them, though; they went on to discover other trees in the forest, and I'm sure you have noticed that, in the branches of those other trees, many of them are a lot closer to the sun than we are. They can see for miles in many directions, but -- ironically -- they still don't know much about our stately and imposing Tree of Mathematics. They know even less about what we are doing in there, huddled by the trunk, in the darkness cast by the thick, obscuring branches. Luckily, they assume we are doing something important. It is, after all, a magnificent tree, and everyone who gazes at its inscrutable glory hopes that someday, somehow, he or she will give birth to a child who can climb it.

Now before I give the impression that I think math teachers spend their lives in the dark, let me remind everyone that I am a math teacher myself. Most of my best friends are math teachers. Also, let me acknowledge that everyone in this room can probably point with fondness to a math teacher in the past who has made a difference in his or her life. However, I dare say that it will be because that teacher taught you about studying, or perseverance, or believing in yourself, or some such enduring lesson of human existence; it will probably not be because that teacher taught you how to rationalize the denominator or how to factor a trinomial -- even though that is what the two of you spent most of your time together doing. You were climbing that trunk, just like everyone else around you was struggling to do, but because you climbed it while looking up at your teacher, you managed to catch a few glimpses of the sky beyond.

The problem is, not everyone on that trunk was looking up. Some were too scared; some became convinced that their arms were simply too short to hug that trunk; still others became discouraged every time they saw how far away they were from the foliage that was to be their goal. Perhaps they couldn't look up; after all, we did focus most of their attention on the finding of roots! Whatever the case, we were scaring away many creative minds, some of whom have since gotten back at us by portraying us negatively in teen-oriented movies. Moreover, we were not getting many of our climbers very far up that tree. I am not here to blame the teachers for this, though; it was definitely not our fault. That's why I'm here to talk about trees.

So let's leave the tedious trunk for a while and talk about the situation further up in the tree, where things are not much better. There, you will recall, everyone is off on a different branch specializing in that one particular twig, virtually unaware of what is happening on the branches elsewhere in the tree. This has created another interesting public relations problem for mathematicians. I am sure you all remember reading not too long ago about the apparent proof of Fermat's Last Theorem, probably the most exciting news story in our lifetime concerning real mathematics. This was to be a very big twig, and the tree was quivering with excitement. It even made the New York Times --twice. But even while being quoted for the record, professional mathematicians acknowledged that only a handful of experts would be able to understand the proof, since, essentially, nobody else was far enough out on that particular branch of the tree. In other words, mathematicians could not explain to reporters the biggest result in their own subject in this century. Fortunately the reporters were accustomed to this, since they spend much of their time dealing with politicians.

This last example, I think, finally illustrates the real problem that we all face in mathematics education today. What has happened is that the tree of Mathematics has grown to the point where it is much too big to know. (Indeed, so have all the other trees in the educational forest, especially the History tree, which grows in real time. But that is another story.) You can know a lot about a branch and everything about a twig, but nobody can know the entire tree -- and we know enough about mathematics to realize that. We forgave ourselves long ago for not knowing all the mathematics, realizing that it would not affect our ability to appreciate, use, and do mathematics. As mathematicians we must be specialists, but we still teach generalists. Unable to teach them about the whole tree, we choose to teach them about the safest part of the tree we know: that sturdy, immutable trunk, which will at least give them the foundation they need for getting up into the branches -- if they can survive the climb. It has also fostered a certain style of teaching in many of us, that style which seeks to cover the necessary material as efficiently as possible, namely the "Here's how you do it. Any questions? Good. Do it." style of teaching. Unless you expose them to the part of the trunk in your lesson plan for the day, you'll never get through the syllabus. There is so much to cover, and so little time. As the tree has grown bigger and bigger, the textbooks have simply grown right along with it, until now we have those seventy-five-dollar, hernia-producing behemoths that are so ridiculously impossible to cover that nobody even tries any more. We realize that the course is inside that textbook somewhere, and we can guide our students through it if we have enough experience on the trunk of the tree, but what do the students think when they see that book? Would you buy a toaster oven if the owner's manual were 600 pages long? Of course not! You would much rather give up toast.

If there is one good thing about the tree getting so enormous, then it is this: We can finally begin to let go of the idea that there is some significant subset of the tree that every educated human being, past, present, and future, should know. This is not an idea which dies easily, to be sure, but I do think that it is useful to question that time-honored assumption. Take, for example, the quadratic formula. I watched Johnny Carson quote that formula from memory during his monologue one evening, to, of course, thunderous applause from an audience of apparent non-mathematicians who recognized it immediately as humorous. He went on to say that he had remembered that formula from high school in Nebraska, and added that his teacher had promised him that he and his classmates would use it later in life. That rash prediction already drew a laugh from the audience, but only because they all knew what was coming. With his usual impeccable timing he rode the swell of that first laugh to its conclusion, then pointed out that he had waited 50 years before finally using that formula for the first time -- to get a laugh in his monologue.

I won't ask how many of you have been forced to make similar promises to your students over the years, but I'd be surprised if you've lasted long in this business without doing so. Just think of how much of your course, whatever it is, is predicated on the assumption that you are preparing your students for future mathematics courses. That is what teaching on the trunk of the tree is all about. Algebra I leads to Geometry, which leads to Algebra II, which leads to Precalculus, which leads to Calculus, which for most students has historically led to the exit. We essentially spend 12 years getting our students ready for Calculus, and when they get there they discover that it is 300 years old, filled with the same calculations they hated in high school, and not exactly worth 12 years of anticipation. So they shinny down off the mathematics tree and strike out into the forest, armed at least with those 12 rich years of valuable mathematical learning: trig identities, the Rational Root Theorem, synthetic division, side-angle-side, FOIL, the Commutative Property of Addition, hey, you name it. Then, the first day on the job out in the Real World, someone notices that they have twelve years of math on their transcript and says with relief, "At last, someone who knows some math! Come here and explain this spreadsheet to me." What will their twelve years do for them then?

Now, I will confess to having fabricated that previous scenario for dramatic rhetorical effect rather than as a reflective argument for revolutionary change. I am not yet inclined to let my students graduate without having studied the quadratic formula. I happen to think that there are good reasons for teaching it, but not because my students will use it later in life. It is, after all, part of the trunk, and I don't want my students to be hanging around the trunk forever. I want them up in the tree. Moreover, there are some other things in the trunk that I am not so fond of, like rationalizing the denominator, and I no longer feel guilty if my students can climb the tree without seeing those. Can that be done? Can students access the tree without climbing up the trunk? Well, the interesting thing, the miraculous thing, the thing that has changed my view of teaching forever, is that yes, now they actually can.

Look around you in the tree of Mathematics today, and you will see some new kids playing around in the branches. They're exploring parts of the tree that have not seen this kind of action in centuries, and they didn't even climb the trunk to get there. You know how they got there? They cheated: they used a ladder. They climbed directly into the branches using a prosthetic extension of their brains known in the Ed Biz as technology. They got up there with graphing calculators. You can argue all you want about whether they deserve to be there, and about whether or not they might fall, but that won't change the fact that they are there, straddled alongside the best trunk-climbers in the tree -- and most of them are glad to be there. Now I ask you: Is that beautiful, or is that bad? Let me warn you that your answer to that beguiling question will probably affect the way you teach for the rest of your lives.

For the record, I think it is beautiful that students of all ages and abilities can access the higher branches of the tree of Mathematics without having to struggle up the trunk. I also think it is healthier for the tree and, ultimately, for the whole educational forest. That is why I plan to spend the rest of my career as a teacher steadying ladders for my students and watching them solve meaningful problems up in the branches. If some of my kids miss part of the trunk or, perish the thought, know less about finding roots, then so be it. Remember: The tree is too big to know anyway -- and I want my students to enjoy the view.

The graphing calculator changed my entire approach to teaching. The first thing I did was let them use it -- all the time. That got me focused on how I would get the students using it, which in turn got me focused on student learning rather than on my own teaching. Then I saw how they worked with each other so well with the calculators, so I began to develop ways to make them work together to discover the mathematics. I now start each class by having them work together on a problem, often the sort of thing I used to use in a lecture to motivate the lesson of the day, only now I wait for them to discover the lesson of the day. Once I saw that they could actually do that, I realized how useless my crisp set of lecture notes had been all those years. Now there is no turning back.

The technology that has made the difference in the tree is, of course, computer technology, but it would never have revolutionized the classroom experience were it not for the fact that it became available in these small, remarkably inexpensive packages. We call this a graphing calculator, but it is actually a computer -- a computer with a very focused mission, running sophisticated internal software that is devoted to mathematics. It does simple mathematics for those with simple tastes, and it does advanced mathematics for those with advanced tastes. More significantly, it also does advanced mathematics for those with simple tastes. A chimpanzee, for example, can produce a perfect graph of y = sin x, while simultaneously clapping his feet with excitement. Most would argue that the chimp will not understand what he has there, and I agree, but some would argue that an Algebra I student would not understand what he has there either, and I disagree. Not only can an Algebra I student understand that it is a function, but he can understand that it is bounded, periodic, continuous, sometimes increasing and sometimes decreasing, with a maximum of 1 and a minimum of -1. He can also understand that the graph changes curvature every time it crosses the x-axis, and with a little explanation he can probably even appreciate that it models harmonic motion. Can he recognize that waves look like that? Of course he can, and if you have an oscilloscope you can prove it to him. Remarkably, he will be able to understand all that without knowing anything about opposite-over-hypotenuse, the unit circle, reference angles, or even what a radian is. He can learn all sorts of things about y = sin x by just playing around on the Tree of Mathematics.

Finally, lest anybody accuse me of not seeing the forest for the trees, let me overwork this arboreal metaphor one more time by applying it to the traditional American curriculum. Our educational forest is very much like the majestic maple forests of Algonquin Park, a place in Canada where I have spent many a summer. It takes centuries for a maple forest to develop, but once its trees are in place, the maples will dominate the landscape forever. Why? Because maple trees drop their leaves every fall, and those leaves eventually form a dense carpet over the forest floor, keeping all but the strongest seedlings from reaching the life-giving soil below. The maples then produce millions of seeds, and theirs are the only seedlings with the strength to pierce through. Maple forests, in other words, have inadvertently evolved a perfect strategy for producing clones of themselves forever.

All the trees in our educational forest are bearing some strong and healthy seedlings. Many of our students leave us and become fine, productive citizens: scientists, teachers, authors, philosophers, doctors, lawyers, mothers, fathers, and even mathematicians. But while our stately academic trees are blooming high above, you might have noticed that not much is happening below to regenerate the forest itself. Look around you: The forest floor is littered with the dead leaves of centuries of curricular material, forming a dense and impenetrable mat that only the strongest young scholars can pierce through. Many of those leaves came from the tree of Mathematics, although the other academic disciplines have certainly contributed their share. Even after the branches of active mathematics have sloughed them off, we keep our leaves around out of respect, or out of tradition, or because they are still in the textbook, or because we are terrified that some teacher in some future course will assume that our students know them and they won't. While it is only a side effect of how trees grow, nothing of deliberately malicious design could ever have been more effective at keeping new trees out of the forest than that litter on the forest floor. The time our students spend with us being educated is very precious; we should not be wasting any of it. Ironically, most good schools encourage all students to take mathematics every year, precisely because they see the aching need for mathematical understanding in order to cope with our increasingly technological society. Little do they realize that we are teaching them the same classical results that we felt their great-grandparents needed in order to cope with the industrial revolution. When do we teach them about the technology that will make the technological society technological? When will they learn what these machines and bigger computers can do? There is already far too much in our curriculum to cover, and the dead leaves just keep accumulating. If the educational forest is ever to be transformed, then I submit that the decay on the floor is the next frontier.

Now that the ladder of technology, in our case the graphing calculator, has demonstrated its effectiveness in getting new students into the trees in their quest for the sunlight, I doubt that the forest will ever be the same. Soon everyone will be buzzing about electronic classrooms, cross-disciplinary learning, multicultural studies, information superhighways, and networking -- curricular concerns that do not fit neatly into the current educational forest. I see them as new holes in the forest canopy that provide wonderful growth opportunities, if only some new trees could take root to take advantage of them. Can we expect some new trees in our educational forest in the near future? Well, nothing is stopping them now but the dead leaves of the Way We Were. The ladder has served us well; now we must bring on the rake.

Not long ago, I attended a meeting hosted by the College Board, at which thirty members of the professional mathematics community gathered to advise the Advanced Placement Calculus Committee on how the AP curriculum should be reformed to conform to the best calculus courses now being offered in our colleges and universities. They didn't always agree, but one thing was for sure: These people came with rakes! The AP Committee listened to their suggestions, then spent several months drafting a new course description for a leaner, livelier Calculus that has now become the AP course we teach.

The NCTM Standards have been in circulation for nearly ten years now, and are actually quite a bit older than that if you count the years of development and discussion that preceded their publication. The changes that have occurred in mathematics education in the last ten years have been remarkable, and although some of them can be attributed to forces other than the Standards, for example technology, the Standards remain the only map that most of us have for walking through the uncharted regions of the educational forest. For that purpose, of course, even the authors realize that the original documents are already inadequate.

It is therefore no surprise that NCTM has been working for some time on their significant update of the original Curriculum and Evaluation Standards to extend the vision of the original document and incorporate what is already known about the new parts of the forest in which many of us find ourselves roaming. It is obvious this time around that the discussions have been much more focused. Speculation about technology has been replaced by facts; theories about student learning have been replaced by case studies; and individual opinions about curriculum content have gradually been replaced by some degree of professional consensus. To paraphrase a mathematical revolutionary of an earlier time, this second version of the Standards sees farther because it has had the luxury of standing on the shoulders of the first.

You may have noticed that while the rakes have been at work beneath the Tree of Mathematics, all their activity has stirred up a pretty good swarm of gadflies. Thoughtful critics, mostly from the university community, have expressed grave concerns about the curricular reforms already in progress, and many of their concerns are quite valid. For example, many are fearful that courses once designed for the more motivated and capable mathematics students will be "dumbed down" to a level suitable for all students. Some fear an abandonment of proof and deductive reasoning. Some fear that students will become dependent on calculators and lose whatever computational skills they might once have had. Of course, none of these outcomes are goals of the Standards, but it appears that the same non-prescriptive approach that has encouraged such a wide variety of reforms has also had the downside of permitting some abuses and misinterpretations. Because the 2000 version of the Standards has had to respond to these issues, we can expect a stronger document to emerge.

Our school, which encourages calculator use from grade seven on, has noted with some regret that even our best students are losing some algebraic facility. While our teachers are still drilling them on fractions, radicals, factoring, and other tree-trunk topics, evidence is mounting that the students themselves are figuring out that these skills are less worthy of their long-term memory cells. Yet even while those skills atrophy, we find that other skills are coming to the fore, such as geometric intuition, creativity, and the willingness to solve a word problem. In other words, I have good news and bad news to report to people on both sides of the reform issue -- which is really what the rake metaphor is all about.