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It's not that being a math teacher is intrinsically bad, although this might be hard for most of you to accept at the start of my story. As graduates of Baylor, I would hope that your math classroom experiences were better than those of most American students, yet even those benign memories have probably been sullied by American culture. You have, after all, been exposed to decades of indoctrination from Hollywood and literature, in which the role of the math teacher has been played by a frizzy-haired nerd in thick spectacles, well-intentioned but totally out of touch with reality. You've all seen these movies, right? The students -- heroes and heroines of the stories -- grow in wisdom thanks to their own creativity and passionate approach to life, while their teachers -- particularly their humdrum old math teachers -- provide the ironic contrast, clinging to their chalkboards, their archaic lesson plans, and their attendance sheets, preaching irrelevant sermons heard only by themselves.

Now, it would be easy to blame Hollywood for this image problem we math teachers have, but I am not here to do that. In fact, I am here to accept the blame myself, on behalf of me and my colleagues across the country. You see, we ourselves have grown into this image, with every good intention, by being dedicated and good at the wrong things. Even we, your super-educated and highly-caring Baylor math teachers, have enjoyed our well-documented successes by doing the wrong things well -- and in the process, we have sold several generations of students on the idea that they can't do math because they don't have it in them. We should never have let this happen, and we apologize, but now we are ready to lead the country out of this educational mess, and we need your help. That is why you must listen to my story. No math teacher can be expected to fight this personal battle alone.

I first realized something was wrong with my profession just after I started teaching at Baylor 22 years ago. I was working long hours perfecting my lectures, making them clear and precise, anticipating every problem that my students might have with the material. One evening I received a call from a distressed parent who was worried that his son was spending far too much time with his homework, apparently because he was not being sufficiently prepared. He noted at one point that his son's teacher the previous year had been so good that his son never even had to open the book, except to get at the homework. Although I felt momentarily chastised, it later occurred to me that an educational model that viewed a closed book as some sort of ideal was somehow seriously flawed. Nonetheless, I worked even harder at making my lectures clearer still, and in time I found myself deriving my greatest satisfaction from students who came back to tell me how terrible their college teachers were, since they could not explain the material as well as I could. Mind you, my former students were lost in their college courses, and they were complimenting me for how well I had prepared them! By that time, I had been teaching long enough that I never even noticed the contradiction.

To understand what went wrong, you have to understand what the education business is all about. For the sake of argument, let's let me represent the education business. We will let this glass of iced tea represent learning. You have come here today to have lunch and to hear a speech, because each of you has the good intention of hoping, perhaps, to learn something. That is what the education business is all about: I talk and you listen. Unfortunately, you have gone about this in the wrong way if learning is what you were after. Remember, learning is being represented by this glass of iced tea.

You know, I like iced tea, but I also like to drink other stuff. A few years ago I was in a local store buying a bottle of other stuff. The proprietor, who knew that I was a math teacher, had made the unfortunate assumption that I might consequently be interested in a discourse on the travails of retail pricing. He therefore told me at some length how difficult it was to arrive at the shelf price of a single bottle of wine. First you took the dealer's case price and divided it by 12. Then there was a mark-up, a fixed percentage that you computed and added on. Then you figured a percentage discount and subtracted that. Then you added on the federal tax, another fixed percentage, then the state tax, then the local tax, and finally you had the shelf price for that bottle of wine. He had been doing this for years. Oh, he said wistfully, what he wouldn't give for a single number that would enable him to do all these things at once! Well, I knew that I had stepped into what we educators call "the teachable moment." Knitting my brow and attempting to look serious, I nodded sagely and said that, yes, I believe I could help him out here. Unsheathing my trusty calculator, I inquired, "Now what were those percentages again?"

In a few moments I had done some first-year algebra on the calculator and was able to give him a single number that would give him his shelf price in the hoped-for single step. When he tried the number out on his $5.89 chardonnay and got the exact price to the penny, I thought the man was going to cry. Before I left that day I had written magic numbers for three different kinds of merchandise on a paper bag, which he then mounted on the wall of his office. I also walked out of the store with a free bottle of champagne.

But here's the point: Do you know how many comparable moments I can point to in my classroom over 22 years, where students have been moved almost to tears by beholding the sheer usefulness of mathematics? Of course you do. Because many of you were in my math classroom, or else one just like it, and you know perfectly well what we were all doing. We were doing "real-world" problems about a man who rows upstream at a constant rate for two hours. Think about that for a moment. Wouldn't you like to meet this masochistic idiot who rows upstream at a constant rate for two hours? Do you know how hard that is for a guy his age? You do know his age, don't you? He is twice as old as Mary, who four years ago was one-third as old as he'll be in ten years. But hey, if you can't figure that out, don't worry; you'll do just fine in life as long as you can remember the quadratic formula.

That, I am afraid, is the math we taught you. We also taught you everything you needed to know in order to compute the magic numbers for wine pricing, but if we had ever put it on the test you would have missed it, unless we had done a "wine problem" before the test to get you ready. Why? Because you knew all along that we would show you how to do whatever irrelevant problems we wanted you to do, and you knew all along that there was a particular way that we expected you to do them, and you knew all along that we already knew the answers. If the truth be known, we both even knew all along exactly which problems you would be able to do and which you would not be able to do, because some kids are just better in math than others. That, too, is what education was all about. We taught you those problems because our teachers taught us those problems, and they taught them because their teachers taught them. We have been teaching the educated citizens of tomorrow with the problems of yesterday, while all the while explaining to them that they will need this mathematics in the future! My friends, this is the stuff of which the emperor's new clothes are made.

Before you get the idea that I am being too rough on math teachers, let me point out that I am a math teacher myself, many of my best friends are math teachers, and besides, this educational model would not have lasted this long if it had been totally ineffective. I dare say that even the English majors at this luncheon can think back fondly about at least one math teacher in your past, and even reflect gratefully on what he or she taught you. However, you will remember things like "discipline" or "learning how to think" or "the value of studying"; you won't remember things like rationalizing the denominator, even though that is precisely the sort of thing that you spent most of your class and homework time doing. Your math teachers taught you to rationalize the denominator in the strong confidence that it would help you one day, but I won't embarrass anyone by asking for a show of hands from those whom it actually helped. The ability to think, on the other hand, helped every person here, and I hope still serves us well today. We have all of our teachers to thank for that, plus our parents, our friends, the media, books, and everyone with whom we have ever shared a conversation.

But thinking has always been a game that people associate with mathematics, so math teachers became particularly distressed about twenty years ago when careful studies began to show that thinking was the one thing that American students were not doing very well. For all the math we were teaching them, they could not use it to solve problems. For all the denominators they could rationalize, they could not interpret the statistics in USA Today. For all the quadratic formulas they could recite, they could not make a decision based on data in a spreadsheet. What could they do? Well, ironically they had a pretty good shot at passing calculus, a 300-year-old subject which was much better suited to the 2200-year-old curriculum that had prepared them. In other words, after 12 years of mathematics classes the American student was specifically prepared for year 13, and that was about it. Moreover, quite a few American students jumped this ship before it ever sailed into the calculus harbor, and wound up being prepared for little of anything. We had met the enemy, and he was us.

The good news is that once the mathematics community had identified the problem, they constituted the perfect think tank to solve it. In the late eighties, with input from every academic society of mathematicians from elementary school through the top research universities, the National Council of Teachers of Mathematics produced this document, Curriculum and Evaluation Standards for School Mathematics, a blueprint for comprehensive educational change in grades K through 12. At about the same time, college and university mathematicians began an ongoing movement to reform the teaching of calculus in grade 12 and beyond. What these two movements have begun is nothing less than an educational revolution, the first really significant change in mathematics education since the Renaissance.

I hear you asking: "What about the New Math?" Let me try to put the New Math into perspective for you. I have collected records since the fifties. Over the years I have invested in more than 2000 45's, hundreds of albums, and high-tech audio equipment on which to play them. When they came out with the 8-track tape, some people claimed that it would redefine the recording industry. I was not impressed, and neither were a lot of other people. The fact was that music was music, and a simple attempt to retool the delivery system was not going to change many people. They also said that the cassette tape would redefine the recording industry, but people kept buying records. They also said that the laser disc would redefine the recording industry, but people kept buying records. They also said that the compact disc would redefine the recording industry, and in an incredibly short period of time, it did. I would never have predicted it twenty years ago, but my lifelong hobby has been redefined forever by a change in the technology.

Which brings me back to mathematics. The New Math never really redefined what we did in the classroom, but I believe now that I have lived long enough to see the compact disc of the math teaching profession. I am holding it here in my hand. It is called a graphing calculator, but it is really a computer, with all the computing power of Baylor's original computer of less than 20 years ago. We require our students to own these from Algebra I on, and they use them all the time. This machine has already redefined the way that we teach and learn mathematics, so it should come as no surprise to you that technology is an integral part of the new Standards for math teaching. Indeed, if you were to point to one thing that separates the educated person of today from the educated person of yesterday, it would be the ability to make use of available technology. Once our students start cruising that information superhighway, that observation can only become more self-evident. As with the CD, there is no turning back now.

When I was a youth, all my radical friends were in reform school. Now all my radical friends are in school reform. It's an exciting time, not unlike the political scene in eastern Europe, and it's really pretty enjoyable once we teachers realize that we don't have to have all the answers. None of my classes begin with me talking anymore; they begin with students working together on a meaningful problem, using books, graphing calculators, notes, and most importantly, each other. When they think they have it solved, I have them compare answers and resolve discrepancies. By the time I put in my two cents, they have already discovered the mathematics themselves and have explained it to each other. I am finally coming to realize how useless my crisp set of lecture notes have been all these years.

It takes a little longer to do mathematics this way, but that is because there are more people doing it. They are, moreover, the right people, the people who are supposed to be learning. If I cover less material, so be it. The fact is, the subject of mathematics has grown so much over the years that there is now actually too much mathematics for anyone to know. Professional mathematicians already realize that there is too much mathematics for them to know, but they don't worry about it, because they realize that it won't affect their ability to understand, appreciate, and use mathematics. They have allowed themselves to become specialists, but they still believe that we ought to be teaching generalists. That is why your child's algebra book weighs ten pounds, costs 75 dollars, and contains more trivia than anyone could ever hope to learn in an academic year of any length. Your child's teacher realizes that the course is in there somewhere, but your child thinks that he has to learn all that stuff in order to be successful. Confronting a kid with that book is probably tantamount to child abuse, but we have been doing it for so long that we hardly think twice about it. The irony is that most of the stuff in those enormous mathematics books is now outdated, since calculators and computers can do it faster and better. There may still be reasons for learning some of the old techniques, but now nobody can tell you that you are learning it because you "will need it in the future."

So now that we have the technology, is there anything left for us to teach your children? Of course there is. Now we can expect them to learn the one vital skill that we failed to teach to most of their parents. Now everybody can learn how to solve the problems that all those rules were invented in order to solve. Moreover, we are now free to let them solve those problems under more realistic conditions. Many of you are business people. If you were your employer, and you needed a problem solved, do you think you would you put your employee at a desk and tell her not to use anything but a pencil and paper, and not to talk to anyone? Would you tell her that you already know the answer, and that you were only asking her to solve it in order to see if she gets the same answer you already have, in the same way that you solved it? Would you tell her not to use a computer, because that would keep her from thinking? Of course not. But that is exactly the way that we math teachers have been testing our students. What we are finally realizing, however, is that our methods of testing themselves have been sending signals to our students about what kinds of performance we value, and those signals have been wrong, wrong, wrong. You do not do your most significant work in 2 1/2 hours. You do not do your most significant work alone. You do not do your most significant work when you repeat what someone has already done for you. You do not do your most significant work when you are surrounded by people who are doing the same thing, especially if you are hoping that they are doing it wrong and you are doing it right. You do not do your most significant work when the best possible result of your efforts is a 2-centimeter oval blackened in with a #2 pencil. Yet that is what we have tested, and our students have gotten the message. Now, even our most successful engineers find that while they can design the machine, nobody can write the owner's manual. It's simply not part of their answer.

I told you earlier that I needed your help. If we are to change education in this country, we need to break away from the idea that students today need to learn the same things, in the same ways, that our teachers tried to teach to us. We grew up in an era when success in mathematics was measured by how well you could imitate what your teacher did, and the best math teachers were the ones who made it the most automatic for their students. Unfortunately, we have been proving for a generation or more that, for most students, that kind of mathematics is a dead end. If your child's teachers are attempting to change their ways in keeping with the new Standards, they need your support, not a lecture about how your old teacher explained things a lot more carefully. In fact, nobody needs a lecture on anything, and I say that as a man who has just delivered one. If you really want to learn about educational reform, my lecture will not suffice; you will need to look into it for yourself, just like I have. Don't say I didn't warn you, though; I said that I was representing the education business. This glass of iced tea represents learning.

As a glimpse of the learning that can go on in a reformed classroom, let me show you how a glass of iced tea can promote real learning. You are the students, and you have come here to learn, but suppose there is no talking head up here like me asking your questions for you. Instead, suppose everyone at the table asks some question that occurs to him or her about the iced tea. We don't have time to collect your questions, so just to illustrate the possibilities I'll break the first rule of classroom reform and give you five of my own:

Admit it: Wouldn't you rather be able to answer those questions than be able to quote the quadratic formula? Are those not the kinds of questions that kids might actually ask about a glass of iced tea? Of course, they would never get to ask those questions in school, because the questions would have to arise separately in classes in geography, mathematics, physics, chemistry, biology, economics, and American history, rather than from curiosity. The kids would then encounter the answers in pieces, each one so far removed from any recognizable context that they would never be able to apply it to something this basic. Think about it, though: If a student could be persuaded to find the answers to just those five questions, imagine how much learning would have occurred!

Incidentally, you probably heard the answers to most of those questions in school, but did you ever really learn them? Even if you did learn them once, was that enough? Would you be able to use your school learning to answer these questions tomorrow for a curious child who might ask? Be careful: You may have learned in school that this tea could have come from Ceylon, and today there's no such place. Social Studies teachers have become quite accustomed to having their courses redefined all the time; now we mathematics teachers will have to learn the same feeling.

Five years from now, if all goes well, American classrooms will be filled with active, questioning, collaborating, conjecturing students, solving meaningful problems with all appropriate technologies. There will probably be far more investigative learning, with questions that cut across several of our historically separate disciplines at once. There will be more emphasis on thinking, analyzing, and creating than on facts, dates, and formulas. From all this will emerge a new model of the educated person, one who can flourish in the information-rich society of the future, that will offer your children more knowledge at the touch of a button than has ever been gathered in any library anywhere.