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It is now ten years since the AP Calculus committee made the historic decision to move the program in the direction of requiring graphing calculators in AP Calculus classrooms and on AP Calculus examinations. The first exams designed for graphing calculator use were not actually administered until 1995, but only because the technology had to first become widely accepted within the AP community. Since then, the AP course description has undergone several major changes to reflect the effects that the technology has had on the teaching and learning of calculus, while the AP examinations have undergone some interesting changes as well. In fact, so much profound change has occurred in the last ten years that it is difficult to keep the significance of this revolution in perspective. After all, resistance to change has been a hallmark of the education profession in virtually every other respect. What happened to make AP Calculus an exception to the rule? I happen to think that it makes for a beautiful story. Moreover, I think that it is time for the story to be told, and I can't imagine a better audience to whom to tell it than the one gathered before me today, since the story of T^3 has been interwoven with the story of AP Calculus reform from the very beginning. So sit back and enjoy a story that I hope you will find both informative and entertaining; it is very much a piece of your own colorful history.

Although the real history of technology in the AP Calculus program begins in the late 1980's, any discussion of calculators in the program has to begin with a little pre-historic event that occurred in 1983. While there were no graphing calculators around at that time, scientific calculators had been around for a decade, setting off prehistoric versions of the current math wars over the role of calculators in mathematics education. NCTM, already aware of many of the issues that would soon be made public in the unsettling document A Nation at Risk: The Imperative for Educational Reform, was convinced that more students would appreciate, learn, and use mathematics if they had access to scientific calculators. The teaching community, however, distrusted calculators, in large part because they were not allowed on the standardized tests that defined success for their students. This persuasive argument focused attention on the College Board's Scholastic Aptitude Test, the national standard for standardized tests at the time.

The College Board brought the calculator issue up with the Educational Testing Service, the group responsible for developing all of their assessments. The ETS statisticians were not opposed to the idea of testing students with calculators, but were understandably nervous about introducing the confounding variable of technology into such a high-volume test as the SAT. With the agreement of their College Board colleagues, they decided to try to introduce calculators into a lower-volume test with higher involvement among professional mathematicians, namely the AP Calculus examination. The AP Calculus committee, convinced that calculators would affect nothing more profound on their examinations than the way students would compute final answers, said fine, bring 'em on, but added that the calculators would be irrelevant for working the problems. Thus it was that scientific calculators were allowed, but not required, on the 1983 AP Calculus examinations.

This was not to be technology's finest hour. Coversations among readers at the 1983 AP reading were replete with stories about papers in which insanity would strike good students toward the ends of problems and cause them to spray 10-digit numbers incoherently across the page. This anecdotal bad news was soon corroborated by the ETS statisticians, whose characteristically thorough analysis of the examinations showed that the calculators, far from irrelevant, had adversely affected such important statistics as the construct validity of the exams. In layman's terms, the calculators were getting in the way of the intent of the examinations, which was to measure a student's knowledge of calculus. No matter why this was happening, this was a fatal psychometric problem, and calculators would obviously have to go. It took another year before the repeal announcement could take effect, and by 1985 the exams were once again technology-free.

This did not, of course, end the conversation about calculators. In fact, it was only a few short years later that the AP Committee was having a routine conversation about a proposed analyze-this-function problem, when committee member John Brunsting raised his hand and changed the course of AP Calculus forever. In his hand was a Casio FX7000 calculator displaying the graph of the function. By 1988, Sharp, Hewlett-Packard, and Texas Instruments had entered the graphing calculator market, and it was apparent to the committee that the teaching and learning of calculus would never be the same. It was time to re-visit the calculator issue — and this time they could not be irrelevant. This time, the tests would have to change. To prepare for that eventuality, the committee began writing "calculator-active" items in 1989, partly to define for themselves what such problems would look like, and partly so that they could be field-tested in advance of their appearance on the AP examinations.

We should take a brief detour here to emphasize what a significant step this was toward the eventual acceptance of graphing calculators by the mathematics education community. Prior to 1989, plenty of discussion had occurred about the pros and cons of allowing students to use calculators. A few forward-looking teachers had even come to realize how effective they could be in communicating mathematical concepts. But when it came to testing, everyone seemed to agree that one of two things must be true: either the calculators did not make a difference, in which case there was no reason to allow them, or else they did make a difference, in which case they obviously should not allow them. The AP committee, informed by their previous experience with scientific calculators, had to work from a contrary assumption. Since they knew full well that technology would change the construct validity of any exam for which technology was irrelevant, the only way to introduce technology fairly was to alter the construct of the exam to make technology relevant. Nobody in 1989 had any experience doing this; indeed, many were convinced that it could not be done without ruining the AP exam. Nonetheless, the committee forged ahead. The first "calculator-active" items they wrote were for scientific calculators, since those were the machines that students had in 1989, but within two years they were also writing "graphing-calculator-active" items — in preparation for a transition that was beginning to look more and more inevitable.

Tom Dick, the current chair of the AP committee has distinguished among three kinds of calculator-active problems. The first is the artificial back-end problem, in which the calculator is simply used as an add-on to perform a final calculation in a problem that could just as easily be done without technology. We wrote quite a few of those in the early days, especially when we were writing for scientific calculators only, although we were not particularly proud of them. The second type is the authentic back-end problem, in which the calculator plays a role in the solution of a problem that would otherwise be unsolvable. Here is an illustration of the difference:

It was not until we started writing problems for the graphing calculator that we discovered the potential for authentic front-end problems. In these problems, the calculator is used at the outset to change representations so that other paths to the solution open up. The following problem is similar to an example we put in the Course Description booklet to prepare teachers for graphing-calculator-active items.

Finding f algebraically by antidifferentiation is obviously not a promising strategy, so we work with

. We can see algebraically that 1 and ð are the only critical number of f on the interval (0, 5), and we can analyze the sign of

to narrow the choices down to (B) and (E). Beyond that, however, the algebraic representation fails us. It is the graph of

, easily produced on the graphing calculator, that enables us to see that f(5) must be less than f(1).

Nowadays it is easy to find calculator-active problems in post-reform textbooks, although the richest repository of such problems continues to be the released portions of old AP exams. It is also the AP exams that have introduced them to the largest audience. More importantly, it is the AP exams that have shown a once-skeptical mathematical community that such problems can enrich, rather than compromise, the assessment of genuine calculus knowledge.

But I digress. Several transparencies ago we left the AP committee busily writing prototype calculator problems. They were also busily preparing two questionnaires, one for high schools and one for colleges, to survey the attitudes of their constituents toward calculators in the classroom and on examinations. By 1990, everything was in place for conducting the Calculator Impact study, one of the most extensive field studies ever funded by the College Board. Nearly 8,000 students from more than 400 schools took experimental multiple-choice examinations with two parts: 20 questions to be answered without calculators, and 10 questions for which a calculator would be either "necessary or advantageous." They and their teachers also filled out short questionnaires about calculator use in their classes. More extensive surveys were sent to 300 mathematics department chairs at the major colleges and universities receiving the greatest number of AP score reports each year. These surveys focused on two issues critical to the committee's impending decision about calculators: (1) the extent to which college calculus teachers were actually using the technology, and (2) whether the decision to require calculators of different specified capabilities would affect their department's willingness to give credit and/or placement to AP Calculus students. An impressive 199 of these 300 surveys were actually filled out and returned.

The College Board then hosted a large gathering of mathematicians from the various professional organizations to consider the data from the Impact Study and to advise the AP committee on how to proceed. The ETS statisticians reported some psychometric concerns that had shown up in the sample exams, but felt that they could be overcome if students had more familiarity with their calculators. The college surveys showed that colleges were not paying much attention to calculators except to disallow them on tests: 24.6% banned all calculators, 44.7% banned graphing calculators, and 50.8% banned symbolic manipulating calculators, although it was not clear that very many of them had ever seen one. As for granting credit or placement if calculators were allowed on AP examinations, fewer than 2% reported that they would change their policy if scientific calculators were allowed, while more than 10% balked at graphing calculators and more than 14% balked at symbolic manipulators. Although the mathematicians who had assembled to advise the AP committee were, as a group, far more bullish on technology than their colleagues who had filled out the surveys, they agreed that a jump to an "any calculator goes" policy would clearly have been premature. In the end, the group advised the committee to announce the requirement of scientific calculators for the 1993 and 1994 tests, but to begin preparing all AP constituents for the requirement of graphing calculators in 1995.

That decision was made public in 1991, kicking off what were surely the most turbulent four years that any AP Test Development Committee has ever had to endure. For starters, nobody was especially thrilled about the requirement of scientific calculators on the exams. Teachers who resisted technology were displeased for obvious reasons, while those who embraced technology had already made the jump to graphing calculators by 1993 and were offended that their students would have to revert to more primitive tools. The committee members themselves, strongly in the latter camp, could hardly defend their own decision (inevitable as it was at the time) and could only plead for patience while hyping their real calculator agenda, which was to bring graphing calculators in by 1995. Ironically, the scientific calculator interim step was so widely perceived as inadequate that it probably hastened the public's readiness to take the next step.

Meanwhile, more trouble was brewing for the committee in the early 90's as they tried to formulate the calculator policy for 1995. They knew by then that they could design the exams, and it was clear that graphing calculators were already making a difference in calculus classrooms across the country. Other graphing calculators, as predicted, had hit the market, and prices had come down. Unfortunately, not all calculators had the same capabilities, and the ones with high-end capabilities still had high-end prices. It was clear to the committee that a serious equity question loomed just around the corner, introducing non-mathematical concerns that were every bit as serious as the mathematical ones. In fact, they were even more serious, as a carefully-designed test could eliminate any actual equity problems, whereas nothing could stop people from complaining about perceived equity problems. In the fall of 1992, the committee decided after much discussion that the only way to eliminate the appearance of equity problems would be to restrict the capabilities of the calculators that could be used. Specifically, they decided that symbolic manipulators and calculators with built-in numerical integration capabilities would have to be forbidden. Although that decision was never announced officially, it was communicated to the calculator companies in a friendly letter. That was when the fertilizer really hit the fan.

Sharp, Casio, and Hewlett-Packard had all entered the market with high-end calculators that did numerical integration. The TI-81, already the top seller in the graphing calculator market, did not. The other companies concluded, perhaps rightfully, that the committee's decision would blatantly favor the TI machines, and they all wrote fiery letters to the College Board to tell them so. John Kenelly and Tom Tucker, former AP committee chairs, also wrote letters to point out how easy it would be for students with TI-81's to cheat the committee's intention by entering integration programs into their calculators. The College Board, bewildered by the committee's reluctance to open the exam to all calculators (and surely more than a little persuaded by the angry letters they had been receiving) sent emissaries to the beleaguered committee to assure them that they should make their decision on the basis of what was best for the calculus program, and to leave the fretting about equity to them. The calculator companies themselves sent emissaries, and in one of the most bizarre meetings I have ever personally attended, told the committee in no uncertain terms that their agenda was to sell calculators, not to make life easy for the AP committee. (The HP representative actually began his presentation with a transparency that proclaimed simply, "Greed is good.") The committee emerged from the entire debacle with a clearer vision of their challenge and with only one notable casualty. Bert Waits, whose incomparable experience with graphing calculators had been of inestimable value to the committee during his short duration as a member, resigned a year early after one of the calculator companies had made his association with TI an issue. In fact, no member of the committee had been more scrupulous about making sure that all calculators were considered than Bert, but, as with equity problems, nothing can stop people from complaining about perceived fairness issues.

Eventually the committee came up with a calculator policy that would make all the calculator companies happy, at least for a while. They would specify the capabilities that a graphing calculator would need to have, and it would be up to the students and their teachers to see that those capabilities were either built into or programmed into their own calculators. This brought most of the high-end (and several more of the low-end) calculators back into the picture. The four capabilities, well-known by now, were: (1) graph a function, (2) solve an equation, (3) compute a numerical derivative at a point, and (4) compute a definite integral numerically. Students could bring in calculators with other capabilities, but could not use them to circumvent the calculus. Palmtop computers, plug-in machines, and machines with QWERTY keyboards were forbidden, more for reasons of security than for reasons of mathematics.

Speaking of security, the biggest hurdle that the new policy had to overcome was the College Board's own strong insistence that all calculator memories would have to be cleared — both going into and coming out of the test — in order to keep the tests secure. The committee was eventually able to convince them that students were not about to waste precious AP exam time typing in the test questions on a tiny alpha-numeric keyboard. If they wanted them that badly, it was easier to write them on their sleeves.

In 1993, even as the first scientific-calculator exams were being administered, the committee began to finalize the plans for introducing graphing calculators on the 1995 exams. Another Impact Study, of lesser scope than the 1990 study, was designed to field-test graphing-calculator-active items, and another survey went out to colleges to determine if the climate was more favorable toward graphers than it had been in 1990. (It was.) Another blue-ribbon panel of external consultants was assembled to advise the committee, and all systems were deemed to be "go." In quick succession, the Test Development Committee made their recommendation, the Mathematical Sciences Advisory Committee approved it, and the College Board trustees passed it. That fall, the historic letter went out to all high schools announcing that graphing calculators would be required on the AP Calculus examinations beginning in 1995.

If the laws of educational inertia had been allowed to determine how graphing calculators were used in AP classrooms, it is doubtful that the AP community could have prepared itself for a change of this magnitude in four short years. History does not suggest that anything in the world of education changes that swiftly. However, the College Board had already established a strong tradition of annual AP workshops for teachers, so all that was needed was for technology to be injected into that workshop network. From 1992 to 1995, a program called TICAP (Technology-Intensive Calculus for Advanced Placement) ran "train the trainer" workshops for AP consultants designed to do just that. This innovative program, led by John Kenelly of Clemson and John Harvey of Wisconsin, was funded by NSF, the College Board, and the major calculator companies, whose generosity put calculators into the right hands: the hands of classroom teachers. The TICAP workshops showed AP consultants the potential of the technology, and the consultants in turn showed thousands of AP teachers. By the fall of 1994, the AP community was ready to begin an adventure in teaching that, at least for many of us, has never really stopped. It is an adventure we celebrate every year at this marvelous conference.

The first graphing-calculator-active examinations in AP Calculus were given right on schedule, in May of 1995. The committee had warned the College Board that lingering public nervousness about graphing calculators might cause a temporary blip in the AP Calculus growth curve but in fact there was an increase in both the first and second derivatives of the growth curve that year. (This anomaly has never been satisfactorily explained.) Meanwhile the AP committee, finally freed from the controversies surrounding calculator implementation, turned their attention to a major overhaul of the AP curriculum. The success of collegiate calculus reform had certainly pointed the way, but it is probably safe to say that technology was the catalyst that had made such reforms possible. Indeed, school mathematics courses in general (and AP Calculus courses in particular) have benefitted from all kinds of curricular and pedagogical reforms that first rode in unnoticed inside the Trojan Horse of graphing calculator technology. The graphing calculator was the paradigm shift; everything else was simply a good idea whose time had come.

The new AP course description became operable in the fall of 1997 and has remained essentially unchanged ever since. The AP calculator policy, on the other hand, has undergone several significant changes to react to the changing technology. The committee had recognized since 1990 that there were advantages to keeping a portion of the multiple-choice test "calculator-free." First, the committee would always be able to ask questions that might otherwise be compromised by technology. Second, colleges who were still nervous about calculators would know that a portion of the grade was earned under traditional conditions. Third, and most important to the ETS statisticians, some data from the test would always be blocked for the calculator variable. The committee has done some fine tuning over the years with respect to the number of problems in each section and the allotment of time, but the multiple-choice split has always been a given. Then, when computer algebra systems became widely available by 1999, the committee decided to split the free-response section as well — a move that was met with gratifyingly little resistance. This further enhanced the credibility of the exam in certain collegiate eyes while avoiding another round of equity concerns based on high-end capabilities like differential equation solvers. At about the same time, the committee felt that it was time to establish a minimal capability for the calculators, namely that the four required operations be built into the calculators rather than programmed in. This condition finally sent the venerable TI-81 into a well-earned retirement.

While it is interesting to recount how graphing calculators made such a sudden and dramatic impact on teaching and learning in American high schools, it is also interesting to recount how computers, which had a ten-year head start, did not. More than ten years have passed since the NCTM Standards presented a vision of the American high school classroom that included a graphing calculator in every student's hand and a computer in every classroom. Of course, the distinction between the two machines was much clearer then than it is today, but most people would still recognize them as distinct levels of technology. While we have come a long way toward realizing the NCTM goal of a graphing calculator in every student's hand, it might be surprising to some that we have made such negligible progress toward that computer in every classroom. I chose the word "negligible" carefully, because even in high schools which are fortunate enough to have a computer in every mathematics classroom, those computers are rarely used to do mathematics. Contrast this with the graphing calculators, which are doing mathematics all the time. Despite their ten-year head start, computers have just not lived up to their potential as players in the secondary mathematics classroom. There are probably many reasons for this, and here are some that occur to me:

First, of course, is the cost factor. Computers are expensive to buy, expensive to maintain, and vulnerable to quick obsolescence. Software, if obtained ethically, is notoriously expensive. Computer glitches, when multiplied by the number of machines required to stock the average American high school, inevitably force schools to hire more personnel to take care of the computers, which escalates costs even further. While plenty of schools have made computer technology an educational priority regardless of cost, there are many more schools for whom all of this expense is simply prohibitive.

Second, a single computer in the classroom usually promotes passive mathematics, while teachers and students need active classroom mathematics. (I am not talking here about computer labs, which pose other problems but certainly not this one.) It is difficult to underestimate the effect of a computer demonstration on a passively observing student, even if the teacher thinks it is pure magic. Students are accustomed to looking at a screen that size and seeing space monsters blow up the White House; do we really expect them to get excited about watching a sequence converge on a spreadsheet?

A third reason that computers have not realized their educational potential in high school mathematics is that these days the high schools seldom see them as tools for doing mathematics. This particular problem extends to computer labs, where geometry teachers seeking to do a little hands-on exploration with their students have always had to wage scheduling wars with classes doing word processing, spreadsheets, games and simulations, and even computer science. Now we have the added complications of e-mail and Internet access, which have the immediate potential of tying up every computer on every high school campus for every available minute. This problem would be tractable if we could make the argument that ours is the more legitimate academic use of the resources, but I would submit that that argument has already been lost. It was lost the day that the computer went on-line. In fact, if anything has the potential to finally justify the existence of a computer in every classroom, it will be the necessity of accessing information from other computers.

I have never worried much that all this technology is being put to uses other than the teaching and learning of mathematics, because I believe that we can have that technology anyway. We can have it right in our hands. Sure, there are things that the larger and more expensive machines can do today that the calculators cannot do, but if we teachers see them as desirable, history teaches us that the calculators will do them tomorrow. Moreover, they will do them more cheaply and more accessibly, and when the next generation of machines comes out it will not cost us $5,000 per unit to replace them. We will have our interactive geometry. We will have our computer algebra systems. We will have things that nobody has thought to ask for yet. If the new machines enhance the teaching and learning of mathematics, they will continue to redefine what we do in our classrooms. And the AP program will adapt accordingly.

Incidentally, I am a supporter of computers; in fact, I was pushing computers in my school long before technology was pedagogically cool. I was part of a three-man committee in 1978 that persuaded our Board of Trustees to sink 100 thousand dollars into a computer system that featured a Data General Nova 830 with 32K RAM, seven CRT terminals, and a teletype printer. Five years later we were back before them, hats in hand, pleading for another 100 thousand dollars to upgrade to a Hewlett-Packard 8000/30, plus ten more terminals and a sensible printer. Five years later we all but abandoned the HP and built a computer lab with shiny new Apple II's, at a cost of another 100 thousand dollars. Those were replaced by an entire multimedia lab, fully stocked with Macintoshes, laser printers, scanners, CD-ROMs, and networking hardware, while our gleeful supplier had another 100 thousand of our school dollars. We are now in the latter stages of another costly transition as the school replaces Macs with PC's. I am gratefully no longer a ringmaster to this particular circus, as our school is well into the personnel phase of the technology commitment, with no fewer than five full-time and three part-time staff members working on hardware, software, networking, and implementation.

The truth of the matter, now so obvious in retrospect, was that we were still teaching and learning mathematics the way we had been doing it for decades. For all its marvelous capabilities, the computer was not changing the way that we taught and learned mathematics at all, and it was costing our school approximately 100 thousand dollars every five years to prove to ourselves that this was so. Convinced that programming was its own reward, we had students writing BASIC programs to do such things as solve quadratic equations. When the emphasis eventually turned to running sophisticated software, mathematics was suddenly just one player in a greater educational game, and a minor player at that, considering the software available at the time. As plenty of other uses for computers evolved to keep our computer labs buzzing, we mathematics teachers virtually abandoned the costly revolution we had fomented and, in many cases, never returned. Today, computer usage continues to gather momentum in every other academic, administrative, and even athletic department, while we high school mathematics teachers can generally be found in our classrooms, content with our graphing calculators. These, at last, are the computers that actually have redefined the way that we teach and learn mathematics.

Four years ago the hot topic in everybody's technology crystal ball was the impact that computer algebra systems would have on the mathematics curriculum. The AP Committee first faced that issue in 1990, since there was already a calculator on the market that had symbolic manipulation capability, namely the HP-48. Fortunately, the syntax was user-unfriendly enough for the committee to conclude that anyone who could get a 5 on the exam using an HP-48 could obviously get a 5 without one. In any event, the potential equity problem it posed was essentially ignored. When the TI-92 came along with the Derive inside, the committee was relieved to see that it had a QWERTY keyboard that automatically disqualified it for use on AP exams. That technicality lasted about a year, by which time all four major calculator companies had CAS built into their high-end, non-QWERTY machines. The committee, well-prepared for this eventuality, responded by putting all these machines on the approved list of AP calculators. Then they promptly announced the non-calculator portion of the free-response section.

What has happened in the two years since then is, I believe, quite fascinating. Despite the efforts of CAS devotees to get teachers excited about the new machines, the response by the teaching community has been strangely lukewarm. Meanwhile the TI-83+, a machine that has basically plateaued except for its Flashed-in extras, has continued to gain ground as the calculator of popular choice. Teachers who have embraced grapher technology every step of the way actually find themselves either ambivalent or uncomfortably resistant when it comes to crossing the CAS threshold. Is this an unexpected turn of events? I think not. If you look at the mathematics journals and professional meeting programs for the past two years, you will find plenty of articles and talks about creative ways to use graphing calculators to teach mathematics, but very few of them have involved computer algebra systems. While mathematics teachers are fascinated by what computer algebra systems can do, it seems that not many of them see CAS as being useful in their primary mission, which is to teach mathematics. I can't say that I blame them, though, because actually, neither have I. In fact, until somebody waves a wand and changes the very essence of what it means to learn algebra, I find it hard to believe that a student with a CAS calculator could possibly be better off than a student without one. Moreover, that probably goes double for calculus.

Here, then, is what I see in my crystal ball: Calculator companies will continue to add bells and whistles as long as it will help them sell calculators; teachers will embrace the bells and whistles as long as they help them to teach mathematics; and students will buy the calculators as long as their teachers tell them to do so. Future generations of calculators will continue to blur the distinction between themselves and computers, but they will continue to have a pedagogical advantage over computers by being cheaper and more focused in their mission. Technology will continue to foster a general move toward theory and applications and away from computational algorithms, just as it has for years. Memorization will continue to decline in importance as a problem-solving tool, which will not sit well with many educators despite the fact that it has never been all that important when we solve real problems outside of the classroom. Meanwhile the reality of the textbook-driven curriculum will guarantee that all educational changes proceed incrementally, providing time for the reflection and peer review that serve this educational community well. The technology we use will evolve symbiotically with the curriculum and will continue to challenge our traditional beliefs about what mathematics should be taught to whom and when.

Whether we will rise to that challenge in a timely manner is quite another issue. If anything has surprised me in the past twelve years of mathematics reform at every level, it has been the almost unquestioned perpetuation of calculus as the unique standard for AP mathematics. As soon as I realized that technology would affect curriculum, I predicted that AP Calculus would eventually need to revert to its original name of "AP Mathematics" because calculus would be too narrow a term to cover the standard first course in collegiate mathematics. This prediction has obviously not come to pass. Meanwhile, what Lynn Steen has called "the teleological influence of calculus" continues to determine what we need to teach to our students in high school, at the expense of many useful, accessible topics in finite and combinatorial mathematics. Students can consequently take four years of high school mathematics and two years of college mathematics and still not know what many of the procedures on their calculators are for. I love and respect calculus, and I have recently acquired a vested interest in its continuation as a textbook for the masses, but it just doesn't seem right that it continues to dominate the general curriculum while so much other worthy mathematics goes widely untaught.

But stay tuned, because curriculum reform is becoming a hot topic in colleges and high schools alike. In fact, it will be interesting to see if the NCTM Standards can succeed in redefining college preparatory mathematics before the colleges bother to redefine what it is preparatory for. One way or the other, something has to give. Meanwhile, just as it always has, the AP program will enjoy the interesting perspective of the monkey in the middle, subject to criticism from above and below, from the left and from the right. Eventually, responding to what it sees, the monkey will move.