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This paper will be more mathematical than political, but I will confess up front that it has a political agenda: I am writing zealously on behalf of mathematics education reform, a subject which has entered a political phase in the collective discourse of our profession. I call it political because there are intelligent, well-intentioned people on both sides of the reform issue, and each side has some difficulty accepting the basic arguments of the other. I, who have seen certain reforms succeed in my own classroom beyond my wildest expectations, feel strongly enough about them to confront the skeptics head on, which I plan to do by analyzing for you the evolutionary forces that had me so set in my own ways for so many years. But make no mistake about it; this is a political paper.

That is why I will begin as any good politician might do, by tugging at your patriotic heartstrings -- in this case with a gratuitous reference to the stirring triumph of our American athletes at the 1996 Summer Olympics! There are many inspirational stories that we could recall from that event with spasms of proud, emotional fervor: the obvious one being the dramatic victory of our young women's gymnastics team. However, I will point instead to the unexpected domination of our young American swimmers, who won twenty-six medals and broke four Olympic records, while competing against athletes whose previous domination of the sport had been so complete that our sportswriters felt compelled to attribute it to chemicals. Who among us was not moved every time the TV cameras showed one of these American heroes wiping a tear away from a freckled cheek, as the stars and stripes rose slowly to the ceiling? What heart did not swell with pride to hear the "Star Spangled Banner" one more time, knowing that it was our country, our son or daughter, our way of life, that stood before the world in victory? Never mind that the Olympics are supposed to transcend those feelings of petty nationalism; we wanted our kids to win. And what, after all, is wrong with that?

Another reason I chose to tug at your heartstrings with swimming rather than gymnastics is that I can understand swimming, because I have done it. I have not done gymnastics. Oh, sure, my school had a pair of those rings that hung from the ceiling, but I had all I could do to simply grab them and hang on for five seconds. Throwing my body into the air and twisting it was out of the question. But when I was a kid, my parents made sure that I learned how to swim.

Reason #1 looks pretty good here, because I was careful to direct your attention to the process of learning how to swim, as opposed to the act of swimming itself. The other reasons are, after all, pretty good motivation for many people at certain times in their lives. People who swim three days a week at an expensive sports club are into reason number two; people who dive for pearls in the Caribbean are into reason number three; people who survive a flash flood are into reason number four; and the coaches of our Olympic team are into reason number five. But you have to be pretty heavily committed to a life of deferred gratification if your motivation for learning how to swim is anything but reason number one.

It is also a happy fact that swimming is not all that difficult to learn. In fact, researchers have noted that babies born in birthing tanks swim instinctively from the moment of their birth, giving a logical purpose to the awkward pumping of all four limbs that appears so uncoordinated when they lie on their backs in their cradles. Very young children can keep themselves afloat with a respectable doggie paddle that they learned genetically from their ancestors. Once they acquire the muscles, they can use that same set of motions to swim. Nobody does of course; doggie paddling is for dweebs. That's why we learn how to swim.

When we learn how to swim, we learn things like the flutter kick, alternate breathing, and the Australian crawl. If we master that, we may go on to frog kicks, dolphin kicks, butterfly strokes, and flip turns. It is not clear that any of this knowledge makes swimming any more fun or useful, but it does enable us to swim faster. In fact, once we have learned how to swim -- something which, remember, we are born being able to do -- it would appear that everything else we learn about the subject is really designed to make us swim faster. If you want to find the reason for doing that on my list, then I believe you will find it at number five. You may not have picked it, but I'll bet you support it.

That last reason, incidentally, is not moot, as there really is an International Mathematics Olympiad, there really are medals, and a talented mathematics student can really win one for his or her country. If you truly want to give reason #5 its due in this context, though, you might want to extend the meaning a bit to encompass all international comparisons between our country and other countries when it comes to mathematics.

Not as much fun this time, is it? Some of you reading this article might be young and idealistic enough to choose #1 as the primary reason for learning mathematics, but I will wager that my older colleagues out there are finding that your evolution as teachers has left you inexorably drawn to reasons further on down my list. People go swimming for fun, but they take mathematics seriously. They excuse themselves for not knowing it, but they still take it seriously. So we take it seriously, too.

Let me explain what I mean by our "evolution as teachers." The typical mathematics teacher enters the profession in the evolutionary phase that I call Phase One: Idealism. We start out with two basic, fundamental assumptions that give meaning and purpose to everything that we foresee about our vocation: (1) that mathematics is inherently fun, and (2) that we can teach it to our students. It is difficult to imagine anyone starting out on the road to mathematics teaching who does not do so under those two assumptions. The fact that they make a lot of sense to Phase One teachers proves that they haven't been teaching much.

A Phase One teacher is a beautiful and vulnerable creature. In this innocent phase you believe that all students can learn. You believe that underachievers can be turned around with the right combination of support and understanding. You believe that, just as all children are born with the innate ability to swim, so are all children born with the innate ability to do mathematics. You believe that you can communicate your own enjoyment of mathematics to all of your students. When thirty students walk into your class on the first day, you feel a personal responsibility for the learning of all thirty of them. You believe that they can learn and you believe that you can teach them. If any of them should fail, you will consider it your failure, too. A Phase One teacher makes a lot of mistakes, but manages to inspire a great many students.

Phase Two comes quickly for some, less quickly for others, but inevitably for all. I call it Doubt. The realities of the classroom accumulate in your consciousness, and you begin to suspect that something is fundamentally wrong with your Phase One assumptions. Either mathematics is not fun, or else there are some students to whom you cannot teach it, or both. Since these assumptions, you will recall, were absolutely basic to your decision to go into teaching, these are very dangerous doubts for a teacher to be harboring, which is why teachers do not stay in Phase Two for very long. They evolve quickly into Phase Three, which is Guilt.

Evolution is almost too benign a metaphor for the Guilt phase, which may in fact be more like a metamorphosis. It is in this phase that you are forced, on behalf of your own sanity, to abandon the Phase One assumptions and pretty much redefine for yourself the nature of mathematics teaching. Faced with a small number of failures that you cannot turn around, faced with some necessary mathematics that you cannot communicate as fun, faced with an occasional angry parent who lets you know that you are the first math teacher that has ever given little Corey less than an A, you begin to feel inadequate for the job. Guilt then drives you to work harder, feeling that if you could only make this course more automatic for the students, if you could only anticipate every possible problem that they might have with the material, then surely you could get them to succeed. When that doesn't work, you feel more guilty, and as a teacher your guilt is magnified by the fact that your adult psyche is bound to the volatile psyches of children, who variously love, hate, fear, and worship you. Their pain is your pain, and considering how greatly they outnumber you, that's not a very good deal for you. It is, of course, at this phase that many mathematics teachers go back to school and become CPA's.

But many teachers survive the Guilt phase and move into Phase Four, which psychologists might call Denial but which I will less harshly call Rationalization -- a good mathematical term. We discover through bitter experience that some students will fail or underachieve despite our best efforts. More significantly, we find that some students do amazingly well for reasons that seem to have nothing to do with our efforts. Smart students can be absent for a week and return knowing more than the students who were in class all along. We discover that some hard-working, lovable kids struggle pitifully with the material, while some lazy, unsavory types absorb it without apparent effort. Some kids just seem destined to succeed in our courses, while others seem destined to fail. And even though some might define success as the mere absence of failure, success is the name of the game. By this time we have moved so far beyond the innocence of Phase One that we have actually adopted a brand new pair of fundamental principles: (1) succeeding in mathematics is rewarding, and most students will succeed if they work hard enough, and (2) if a student can't succeed in mathematics by working hard enough, then for that student, mathematics will never be rewarding.

These two principles look pretty good, but notice that we have come a long way from Phase One. There are now conditions riding on the enjoyment of mathematics, and those conditions are based on success rather than on the mathematics itself. If you think about that for a minute, you realize that this ought to put a very high premium on the way we measure success! I intend to return to that point a little later, but first let's take a closer look at that second principle. Not only does it subjugate the enjoyment of mathematics to the goal of success through hard work, but it also admits the existence of students who are inevitably out of the loop: not lazy, unmotivated students, but students who can't succeed by working hard enough -- sincere students for whom our inherently difficult course will not be rewarding. To preserve the integrity of the mathematics and the dignity of the students, we conclude that these students should be separated from the mathematics. They are the students who are wonderful human beings in every way but who, and I quote, "are not able to succeed at this level of mathematics," or else students who are wonderful human beings in every way but who, and I quote again, "have not had an adequate background in mathematics." Notice that, in either case, these are wonderful human beings in every way who are destined for unhappiness through no fault of their own or ours. Therefore, since we care for them as human beings, not only should we forgive ourselves for not teaching them if they should show up in our classes, but we are actually obligated to counsel them the heck out of there before they make a terrible, avoidable mistake that will render life difficult for us both.

A mathematics teacher can pretty much spend the remainder of his or her professional career in Phase Four, and many do. Occasionally some teachers will escape into Administration, but this just provides them with a power base from which they can institutionalize the two new principles that they acquired in the Rationalization phase. Now they can tinker with the curriculum and design alternative courses for the students who can't do mathematics, thereby giving every student the chance to succeed at his or her own level. The top students, those who are good at mathematics, can do their thing in the hard courses without interference, while the struggling students can be passed on down the line to courses that have less mathematics. Soon their schools will be so comfortably locked into Phase Four that even their fresh, young teachers will react to a student's cry for help by first looking for a lower section to which that student could be moved. Administrators also get to explain this caste system to parents, most of whom will accept it completely despite the fact that their children's education is at stake. The notion that success depends on hard work is easy to sell; after all, that is the American way. The notion that not everyone can achieve success in mathematics should be harder to sell, except that people find it strangely comforting -- particularly those who have always believed themselves to be in that category.

Let me pause here and assure you that I am not claiming that all students can do mathematics equally well, nor that all students will enjoy all mathematics. What I am claiming is far more ludicrous than that. I am claiming that our evolution as teachers has created a system wherein the responsibility for convincing students that they cannot succeed in mathematics, and that they will consequently not enjoy mathematics, lies with the teachers of mathematics! Incidentally, we mathematics teachers seem to be the only ones afflicted with this paradoxical mission. English teachers may fail students for all kinds of reasons, but they never tell them that they weren't cut out for English. History teachers don't tell students that they can't study the Civil War because their background is weak in Western Civ. Chemistry teachers don't tell students that they are doomed because they can't do chemistry -- although, curiously, they do sometimes tell them that they are doomed because they can't do math. Only mathematics teachers, it seems, feel responsible for shielding students from failure by identifying our failures before they can occur. It is the Phase Four solution.

Fortunately for our profession, though, there is life after Phase Four. If you teach mathematics long enough and acquire enough experience and perspective, you may begin to enter another idealistic phase, wherein the original Phase One assumptions begin to look reasonable once again. I call it Phase Five: Rejuvenation. No single event shakes you out of your Phase Four coma, but rather a multitude of little events. You meet a former C student who sleepwalked through your course and today is making $120,000 a year in the software industry. You meet a former A student who did everything right for you, then went to college and never took another math course. A girl in a lower section of senior math tells you that "she never could do math," and you suddenly realize that she learned that idea right there in your school, on her way down the ladder to that lower section. You find your slide rule in your desk and it reminds you of how many hours you spent in school studying things that turned out to be pretty useless, even in mathematics. You pick up your 600-page algebra text and ask yourself the intriguing question, "How many important ideas are really in this book, and how many students are really incapable of understanding them?" These are little events but they add up, until one day you look at yourself in the mirror and you see a Phase Four teacher wearing the Emperor's new clothes. You had come into this profession to teach students, and now you realize that you have unintentionally let a whole lot of them get away.

So you enter Phase Five. You walk into your classroom in September and resolve that everyone in that room will learn mathematics from you that year, because they are your students and you are their teacher. They will enjoy it, too, because you enjoy mathematics, and you will communicate that enjoyment to them. Nothing will stand in your way: grades, course outlines, counselors, parents, administrators, textbooks -- nothing. You realize that it is all Phase Four, designed to weed out the losers, and you have been there, done that. You are Phase Five, by thunder, and you rode into town to teach mathematics!

You will, of course, have to overcome the fundamental hypothesis of the Phase Four universe -- that pervasive notion that success is the name of the game. Here is where you come to the obvious, but radical, conclusion that to achieve your ultimate goal -- the teaching of mathematics -- you just might have to redefine success! It sounds simple enough, but it is in taking that small step that you will cross the threshold into a world which will make perfect sense to you, but which your colleagues might well interpret as the onset of your senility.

From the Phase Five perspective you see that there are many, many problems with the way that we have historically measured success in mathematics, not the least of which is that what we measure has virtually nothing to do with success. Open your test file and take out a typical test; it probably doesn't matter which one. Now imagine a student who gets 50% of that test right. The first question you need to ask is, "Who cares?" The answer, of course, is that a lot of people care: the student, the parents, the college admissions offices, your department chair -- all of them care very deeply, because that is the way that you measure success, and success is the only goal they understand. But what kind of success does that test measure? What kind of pride should a 92 on that test bring? In ten years will the 92 student remember how to answer any question on it? What has been accomplished, and is it really mathematics? Should that 50% student fail, and if so, what has he failed to do? Can anything good come out of that test at all? The only possible rationale for a test in the first place is to promote the learning of real mathematics, and the connection between most of our tests and real mathematics is tenuous at best. Yet it would only take two or three of those tests to convince some students that they are not cut out to do mathematics. They will drop the course or move to a slower section, and our chance to teach them mathematics will have been lost, perhaps forever.

A full discussion of the shortcomings of our traditional assessments would fill another entire article, but for the purposes of this paper let me make one simple observation: If a sincere student gets 50% of one of my tests right, I will not feel compelled to give him a 50 — because I can redefine success by scaling my grades. If I want to, I can scale that 50 to a 73 (assuming that I scale the rest of the class equitably). Sure, I have colleagues who will call me Dr. Feelgood and claim that I am sacrificing the integrity of mathematics, but a 73 keeps the student in the fight, while a 50 knocks him out in an early round. Forget self-esteem; which strategy would you say promotes the learning of mathematics? And let's not kid ourselves that a student who gets an 92 on my logarithm test won't mess up the laws of logarithms in a calculus course someday. You can spend eight weeks on this stuff, and your best student will still do this in calculus a year later:

If I could return for a moment to my swimming analogy, think for a moment how you might assess success among swimmers. How would you determine competence? Excellence? Which people would you label as "non-swimmers"? Most of us would probably define competence pretty broadly, something like the ability to do whatever it takes to stay afloat in deep water under reasonable conditions. Excellence would probably be measured in terms that have surprisingly little to do with competence and a whole lot to do with skills: how many strokes you can do well, how fast you are in the water, your breathing rhythm, and so on. But what would you say about a pearl diver who knows only a few basic strokes, but who swims all day for a living? Is that person excellent, or merely competent? What about someone who has a mask, a snorkel, and fins on his feet? (Technology!) Should we even count that as swimming? If you take your kids to the lake, can they have a good time swimming if the form is off on their flutter kick? And does that count for anything, especially if the object of going to the lake was to swim? Fortunately for all of us who love the water, success in swimming can be interpreted in many legitimate ways. It would be nice if tthat were also true in mathematics.

The fact that it is not true in mathematics is the final chapter in my story. We can't really blame that on the evolution of us individual teachers, because we define success in mathematics the way that the system defines it for us; that is, we teach our students what is in the curriculum, and we model our test exercises after those that are in the textbooks. Here is where I get to quote one of my favorite scientific aphorisms of all time: Ontogeny recapitulates phylogeny. [1] This is apparently a pithier version of an assertion made by the nineteenth-century biologist Ernst Heinrich Haeckel, the man who also first coined the term "ecology."

Ontogeny refers to the evolutionary development of the individual, while phylogeny refers to the evolutionary development of the group to which the individual belongs. The recapitulation part is manifested in such curious phenomena as the temporary appearance of gills and a tail in our own embryonic development as humans. In other words, the evolutionary path we follow as individuals parallels the evolutionary path we followed as a species. This idea has apparently been discredited by subsequent biologists, but I intend to borrow it anyway to extend my evolutionary metaphor in mathematics.

As mathematics teachers, we have followed an evolutionary path that has recapitulated the evolution of mathematics education in general, at least in our American schools. Phase One, symbolized by the little red schoolhouse, saw mathematics as something useful to be taught, like reading and writing. Everyone learned it, and it had immediate practical value. Phase Two came into being around the beginning of this century, when the college community decided to teach all students a college-preparatory curriculum that would include algebra and Euclidean geometry. Phase Three followed shortly thereafter, when large numbers of students decided that they hated algebra and geometry. Phase Four came when the system adapted to the non-mathematically-inclined by essentially writing them off, allowing them to take other options instead. Then, in 1958, the Russians put a grapefruit-sized piece of hardware called Sputnik into orbit, and suddenly success in mathematics became a matter of national priority.

The system was well entrenched in Phase Four by that time anyway, but the Space Race focused the attention of the academic community intently on the curriculum that the best and the brightest would take. The important thing was to prepare the mathematically talented students for rigorous university courses in calculus and science. If it took a rocket scientist to understand the courses, all the better; rocket scientists were exactly what we needed! Mathematics became a competitive enterprise, and those who succeeded were more isolated than ever from those who did not. Algebra and analysis courses moved away from genuine problem solving and toward an axiomatic approach that did little to motivate curious minds. But that was okay; in the Phase Four way of thinking, it was sufficient motivation that they would need it later. Mathematics courses thus became exercises in deferred gratification, each one a preparation for the next, with a payoff at the end for those who were strong enough to survive. That was success in mathematics, and to mess with it would have been tantamount to putting our nation at risk. In 1983 a national commission discovered the educational disaster that all this Phase Four tinkering had created. They called their report "A Nation at Risk."

It seems that the down side of a system that encourages those who can succeed and that weeds out those who cannot is this: You get some great Olympic swimmers and a lot of people who can't swim. That was the situation in mathematics when "A Nation at Risk" came out and rattled our Phase Four cages. That document reminded the country that mathematics was in fact, important for all students to learn, just as we mathematics teachers had been saying all along, but it went on to point out that shockingly few students were learning it. Moreover, the mathematics being taught was not the mathematics that the students needed in order to cope with the modern world. The stage had been set for the Rejuvenation phase of our phylogeny, currently being orchestrated by the NCTM Standards and by the "calculus reformers." Like most evolutionary changes it can not be expected to take place overnight, but if you believe in natural selection at all, you suspect that it inevitably will.

Right now we are caught in a temporarily uncomfortable transition period as teachers. While each of us can be in any of the five phases of individual development, our profession is evolving from Phase Four to Phase Five. As that evolution takes its sweet time to occur, you will probably notice that it is difficult to get a single expert opinion on anything; and as for consensus: that is virtually impossible. You can easily tell that I am a believer in education reform. I have tried things in my classroom that I would have laughed at as educationist nonsense ten years ago, and I have seen my students flourish mightily as a result. But I spent two decades in Phase Four, and I know where my respected colleagues are coming from when they criticize the Standards as minimalist fluff, the fad of the day, MTV Math, New New Math, or simply the death of American education. These critics succeeded in math, just as most of you and I did, in a Phase Four world, where the curriculum was tailored for those who could succeed in mathematics, and where that success -- and this is the important point -- was defined by what you could become, not so much by what you were actually doing. Their criticism stems from a genuine concern for the future of our students. What will they become? I can show the skeptics my classroom today, where students are actively engaged in doing mathematics for the entire period, talking, conjecturing, arguing with partners, communicating solutions at the board -- and none of this will convince them if they can find a single computation in the textbook that one of my best students cannot do. Never mind that you could have found such a problem in any textbook for any class I have ever taught -- or, for that matter, for any class I have ever taken -- as long as I was at least trying to teach them every conceivable basic skill, I was playing my appropriate role. If I would just stick to that role, then, somewhere down the line, the few students for whom my teaching was really intended would blossom and begin doing real mathematics.

But those who have actually read the Standards realize that the document calls for all of our students to be doing real mathematics all the time. The Phase Four reaction to that goal is predictable: If all students can do it, then how can it be real mathematics? So they conclude that the Standards must call for a dumbing down of the curriculum, so that students would do less mathematics. In fact, the opposite is true. According to the Standards, the students should be doing more mathematics; it is we who should be doing less.

It would be nice to let all this controversy pass us by, but how can we? These are the issues that define what we do as teachers! So I propose that you not wait for the phylogeny to settle out, but that you concentrate on ontogeny instead. If you have made it to Phase Four, you can make the transition to Phase Five yourself and wait for the rest of your colleagues to catch up. Trust me: It's much less stressful to watch this evolution from the place you're going than it is from the place you're leaving.

Remember the ideals you had when you began as a Phase One teacher. Remember that you enjoyed mathematics, and that you once believed that you could communicate that enjoyment to any student who came your way. Remember the disillusionment you felt in your Doubt Phase and the guilt that you felt in your Guilt Phase when you discovered that you were not reaching some students. Reflect on the paradox of a Phase Four system that proclaims the importance of mathematics for all students, then sets out systematically to weed out those who fall short. Analyze realistically the importance of the curriculum you have been teaching and the significance of the mathematics you have been testing. Then read the Standards and decide for yourself which vision of American education consists of more mathematics being done and which consists of less.

I am not a flaming iconoclast out to destroy our graduate schools of mathematics, nor are the distinguished colleagues of my profession who are calling for reform at every level. Someone who wants everyone to learn how to swim is not out to sabotage our Olympic swimming program. Indeed, it would seem that a country that swims actively, enjoys swimming, and appreciates excellence in swimming is likely to produce better Olympic swimmers than a country where people claim openly not to understand swimming and where swimmers themselves are considered by non-swimmers to be geeks.

We still need to ensure that our best students have every opportunity to hone their mathematical skills, but we also must encourage them to expand their horizons, exercise their creativity, and learn how to communicate -- things that have been sacrificed in the Phase Four classroom in order to accommodate more content being covered. We also know more about learning now, and it appears that while we were teaching rote skills to good students so that they could do meaningful mathematics later, they could have been doing meaningful mathematics all along and picking up the skills along the way. I believe that the latter approach has the potential of delivering more and better candidates to our colleges and universities for further mathematical study.

Everyone loves gold medals, and everyone wants our American athletes to win them. In swimming we can train Olympic athletes by identifying them young, drilling them on fundamentals, and putting them through the kind of rigorous training that would kill most recreational swimmers. That kind of swimming is not for everyone, and yet there are plenty of good reasons for everyone to learn how to swim. And we do not have to restrict non-Olympians to the doggie paddle, either; we can teach everyone how to swim confidently and competently, with all the enjoyment and health benefits that come with the sport. So, if we can do that with swimming, why not with mathematics? Remember, both swimming and mathematics can be learned and enjoyed starting at a very early age.

We did not get into this profession just to teach future world-class mathematicians, nor did our schools put so much mathematics into the curriculum so that only a few students could really learn it. Students, parents, colleges, and employers know that mathematical thinking is important for every student, and that it will probably become more important as time goes on. We should know that better than anyone. Let's listen to the Phase Five reformers who are calling us back to our original mission as teachers. Let's reconsider what kind of mathematics is important to be taught, and let's see if we can't teach it to everyone. Let's get to work on creating a future in which the great mathematicians, just like the great swimmers, are simply the best at doing something that everyone else can do. The only thing that stands in the way of that future is the evolutionary burden of our past.