THE TEST DEVELOPMENT COMMITTEE
Ben Klein - Davidson College - Davidson, North Carolina

Editor’s Note: Dr. Benjamin Klein is a professor of mathematics at Davidson College, a member of NCAAPMT, and a member of the Test Development Committee for advanced placement calculus. In this article, Dr. Klein writes about the work of the Committee.

When I joined the Committee, several members told me that their work on the Committee had been one of the most rewarding professional activities they had ever taken part in. Now that I have enjoyed a full year's experience on the Committee, I agree with this assessment. Working with good people, both the other members of the Committee and the Educational Testing Service (ETS) staff, on a project with substantive mathematical and pedagogical aspects has really been a treat.

For the record, the Committee currently consists of four college faculty and three secondary school faculty. However, the Chief Faculty Consultant, Larry Riddle, and at least one ETS staff person always attend the Committee's meetings. The ETS staff members provide virtually all of the administrative support that the Committee requires, and the actual physical preparation and distribution of the examinations is an ETS responsibility. The Chief Faculty Consultant helps the Committee assess the "gradability" of the items it considers. Interestingly, the Committee's members are paid by and answerable to The College Board, not ETS. The Chief Faculty Consultant is paid by ETS. In practice, the Committee functions as a single unit and no one pays any attention to who signs whose paychecks.

The Committee is responsible for both the actual Advanced Placement AB and BC examinations and the course descriptions from which these examinations are drawn. In the paragraphs below, I will concentrate on the examinations, since the Committee is agreed that, given the wholesale changes that were made in the course descriptions a few years ago, it is too early to make much in the way of substantive changes in the course descriptions now. However, in its October 2000 meeting, the Committee did decide to move slope fields from the BC to the AB course topical outline, effective with the 2003-2004 academic year. A discussion of the Committee's rationale for this decision appears else-where in this issue. When I joined the Committee in the fall of 1999, the other college members were Tom Dick, Committee Chairman, Stella Ashford and David Bressoud. The secondary school members were Mark Howell, Martha Montgomery and Nancy Stephenson. Like me, Nancy and David were rookies, little knowing what we were in for. It is worth noting that it was unusual to have three new members join the Committee at the same time. Since Committee members are appointed for three or four year terms, the normal turnover is two members per year.

At the end of the 1999-2000 academic year, Mark and Martha rotated off the Committee. If you know either of them, you can imagine how valuable their contributions to the Committee were. Their able successors on the Committee are Maria Perez Randle and Mike White, who attended their first committee meeting in October of last year. Maria and Mike bring their own strengths and helpful new perspectives to the Committee.

In an effort to give some insight into what the Committee actually does, I will give below a brief discussion of the timetable employed by the Committee as it carries out its charge to develop four calculus examinations each year. I say four examinations because both the AB and the BC examination are offered in operational (regular) and alternate versions, the later taken by students who, for valid reasons, cannot take the operational examination.

My comments below focus on the 2002 examinations since the Committee has just adopted a new schedule for its meetings. During my first year on the Committee, we met three times. However, in recognition of the effort required to write four new examinations each year, the Committee has added a meeting to its schedule and will meet four, rather than three, times a year. Effective with the 2000-2001 academic year, Committee meetings will typically be held in October, January, March and July. The Committee did meet four times a year during the years in which the curriculum was undergoing major revisions, but that period predates my tenure on the Committee.

In May 2002, perhaps as many as 200,000 students will take one of four Advanced Placement Calculus Examinations. The Committee started work on the free response portions of these examinations almost two years earlier, in October 2000. In some sense, the work started even earlier than that when individual members of the Committee drafted test items for the Committee's so-called "free response pool." During the October 2000 meeting, we identified the areas we wanted to test on the free response section of the regular AB and BC examinations, and then we went to the pool to try to find problems that fit into these areas. In some rare cases, we found problems that, as written, were pretty much what we were looking for. These needed only minor "tweaking," a word, by the way, that is frequently heard during Committee meetings. More typically, we found problems that met our needs only after a good bit of work in both editorial and mathematical requirements. In some cases, nothing in the pool came even close to meeting our needs, in which case, one or more members of the Committee volunteered to draft items that did.

By the end of the October meeting, we had a pretty good idea what the free response items would be. We worked individually on these problems with the intent of having final drafts ready for review at the January meeting. Then at the January meeting we reviewed the drafts for the operational examinations and started the process of putting together the alternate examinations, using procedures much like those we used in October. By the time the Committee meets in March, we will have final drafts of the free response items for all four examinations ready for review. These will be reviewed again during both the July and October meetings, the latter with several new sets of eyes as new members join the Committee. The expectation is that the 2002 free response items will be finalized at the October 2001 meeting, culminating a development period that spanned a full twelve months.

Just as Committee members write questions for the free response pool, members write items for a multiple-choice pool. Items in the pool are reviewed by the entire Committee during its meetings. Each item is either approved, sometimes with modifications approved by the full Committee, or rejected. The ETS staff selects, from the list of approved items, the problems that will actually make up the multiple-choice sections of the four examinations. A first, preliminary draft of these sections will be presented for review at the March 2001 meeting of the Committee, and, based on the Committee's feedback then, that draft will be revised and reviewed again in July 2001. Then, a final draft will be presented for approval at the October 2001 meeting. Thus, with the exception of minor editorial changes, each of the four examinations for 2002, in its entirety, will be ready to go to press in October 2001. For someone, like me, who is used to writing a test the night before it will be administered and then photocopying it the next morning, having anything ready that far in advance seems unnatural indeed.

It should be clear that the timetable outlined above provides multiple opportunities for review of the questions that will constitute the four examinations. I need to emphasize that the items are reviewed not only during the actual meetings but also, individually, between meetings. This is just one manifestation of the great care that is taken on the part of the Committee and ETS to get the examinations "right."

The Committee is very much concerned with issues of equity and does everything possible to make the examinations fair. We try to avoid questions that have a gender or racial bias. The experience of the ETS staff is invaluable in this regard. In recent years, we have been much concerned with issues of calculator or technological equity, specifically, the so-called C(omputer) A(lgebra) S(ystem) advantage. The TI-89 is one of several calculator models that offers this advantage, and we are very much aware of the CAS prowess of the TI-89, for example, relative to other calculators. We have to be careful to write questions on the calculator active sections that do not give a demonstrable preference to students using a calculator offering a CAS advantage. Indeed, the very existence of the non-calculator sections on the AP examinations is a product of the introduction of calculators with significant CAS features, with the consequent and dramatic non-leveling of the computational playing field. [Cf. pages 15-16 of the May 2000, May 2001 Course Description.] The introduction of the non-calculator sections is not at all, as some may imagine, an attempt by the Committee to move the examinations back to the days of purely paper and pencil based computation.

To give some feel for what is involved in the CAS advantage issue, consider the following, interesting problem on which student with a calculator like the TI-89 would have a huge advantage.

Find the positive real number such that the area under one arch of the curve and above the x-axis equals 1.

A student with a TI-83, for example, really needs to do both some calculus and some algebraic computation here. A student whose calculator is CAS capable, needs only to realize the connection between areas and integrals, do a little mental trigonometry and then type in, using the right syntax, something like the following.

Solve(integrate)

The calculator obligingly spits back the desired . The conclusion drawn from this exercise is that the problem in question needs to be relegated to a non-calculator section of the examination.

Contrast the problem above with the following calculator active problem which is fully accessible to a student with a calculator that can solve equations numerically and numerically calculate the value of a definite integral, these being two of the four required calculator capabilities.

Find the area of the region in the first quadrant that is bounded above by the graph of and below by the graph of . This time there is no CAS advantage, and this problem might well appear on one of the calculator active sections of an AP examination.

As we all know, the introduction of the first graphing calculators forced the mathematical community to reassess both the content and the pedagogy associated with AP Calculus, with resultant changes in the AP examinations. The example above makes clear that the introduction of graphing calculators with CAS adds one more dimension to the construction of the AP calculus examinations.

The Committee wants to keep lines of communication open with both AP teachers and with college departments of mathematics. Thus, during the AP reading, there is always an evening session during which readers can address the Committee with concerns and questions. Then, once each year at a teachers conference, The College Board provides an opportunity for interested parties to meet with the Committee. In this spirit, I encourage any one who wants to contact me directly about the Committee and its work to simply write (Department of Mathematics, Davidson College, POB 1719, Davidson, NC 28036), e-mail (beklein@davidson.edu) or call (704-894-2318) at their convenience.

SLOPE FIELDS ARE COMING!
Earl Mitchelle -  The Asheville School
Asheville, NC

College Board announced last November that slope fields will be added to the outline for the AB examination beginning in the 2003-2004 academic year and will be effective for the 2004 examinations. In the Advanced Placement Program Course Description (Acorn book), In the Topical Outline for Calculus AB, II. Derivatives, Applications of derivatives., the following will be added.

“Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solutions curves for differential equations.”

Recently on ap-calculus@list.collegeboard.org, an online discussion site, several people asked why Euler’s method was not added to the AB outline at the same time that slope fields was added since it is a topic about differential equations that is associated with slope fields in the BC course description.

Ben Klein, a member of the Test Development Committee explained, “Slope fields provide a nice geometric visualization for not only differential equations but also for some introductory comments about integration. There are colleges courses (Calculus I) that introduce and use slope fields so the topic can be viewed as one belonging in the first semester. On the other hand, Euler’s method does not provide the same level of insight. Thus, since we are reluctant to add topics to AB without deleting topics, and there is nothing we want to delete right now, we decided to minimize the number of changes we made. It’s also worth noting that we need to maintain an adequate number of BC only topics.”

Slope fields was a topic in question BC4 on the 1998 examination and question BC6 on the 2000 examination. Question BC4 on the 1998 examination gives the differential equation . Part (a) calls for a sketch of a slope field on a given coordinate system with points at (-1, 1), (-1, 2), (-1, 3), (0, 1), (0, 2), (0, 3), (1, 1), (1, 2), and (1,3). Part (b) asks the student to find the particular solution for the given differential equation with the initial condition . Part (c) requires the application of the technique of separation of variables for solving differential equations which is in the AB course outline.

Question BC6 on the 2000 examination gives the differential equation . Part (a) is a slope field with the points      (-2, 1), (-1, -1), (-1, 0), (-1, 1), (0, -1), (0, 0), (0, 1), (1, -1), (1, 0), (1, 1), and (2, 0). Part (c) asks for the particular solution to the differential equation with the initial condition .

Complete solutions for BC4 (1998) and BC6 (2000) can be found at the College Board web site for AP calculus at http://www.collegeboard.org/ap/calculus. Click on 1998 Free-Response Questions and 2000 Free-Response Questions.

It is reasonable to expect that a question combining slope fields and separation of variables could appear on the 2004 AB examination. AB teachers might consider adding slope fields to their courses before the 2003-2004 academic year. Although Euler’s method is not a topic in the current AB course description and is not going to be a topic in the AB course description in the near future, a unit on separation of variables, slope fields, and Euler’s method would be a good addition to individual syllabi for AB courses.

PROPOSAL TO CHANGE GRADE WEIGHTING SYSTEM
IN NORTH CAROLINA
Jane R. Barnett - Scotland High School
Lourinburg, NC

There is a proposal before the Education Cabinet to the North Carolina Legislature to sharply reduce the way grades are weighted for Honors and AP courses in computing a student’s GPA and in determining class rank in schools across North Carolina. I would like to address the prospect of making these changes. The reasons weighted grades have been used over the years seem to have been lost in the proposal.

Since young people often choose the path of least resistance, taking a more rigorous course of study is an easy choice for only a small percentage of high school students. Many others who are very capable, or at least have the ability to handle an accelerated pace and more challenging curriculum, need some incentive to take these courses.

Scholarships and college admission hinge on GPA and class rank, so college bound students must weigh the valuable preparation provided by the more difficult classes with the possibility that they might make a B or a C and jeopardize their class rank. Finding that those with the highest GPA have taken the easier versions of the academic courses and electives where they can more easily make an A, would discourage students from taking Pre-Calculus rather than Algebra III, or English 11 Honors rather than regular English 11.

We often say that we want high standards and increased student achievement. Since many would be unable to perform at the level an honors class requires, there must be more basic classes available. However, those who would benefit more from a challenging honors version, but want to have a higher GPA, would choose the basic course. The student will be the loser, and less demand might even cause schools to stop offering the honors classes.

We say we want to narrow the achievement gap for minority students. Again, the weight-ed grades are an incentive to risk taking harder classes. Some students whose friends might not take college prep classes need encouragement to work to their potential.

A student can be appropriately placed in an honors class and still not make an A. There is a range of grades in most classes. We want students to take the most difficult courses in which they can be successful.

The proposal that only classes with End-Of-Course (EOC) tests be given honors weight, and then only for those making a top score on the EOC, makes the additional quality point a reward for performance on one multiple choice test. Historically, the additional quality point has been an adjustment to acknowledge the added requirements and rigor for all students taking the class. The A student would already have the highest average. With this change, only they might get an additional boost for taking an honors class (5 QP’s), thus widening the gap. No B students need take an honors course. They might make a C on that, and fall even farther behind (2 QP’s).

The as yet undetermined grade needed on the EOC test to earn the QP has been mentioned as the 90th percentile (on the scale used to determine eligibility for Governors’ School, not the suggested grade on the exam). This would mean only a small number of students would qualify for the weighted grade. Those taking courses without EOC tests, such as Pre-Calculus, English 11 Honors, and honors courses in the arts and in foreign languages, would not be eligible.

The proposal that AP students get one point on the grade earned in the class and another only if they take the AP exam and score well on it would only benefit underclassmen. The scores aren’t known until July, so GPA and class rank are no longer computed or requested for seniors. Also, the point would be a reward for performance. College credit is the best reward for a good AP exam score. The difficulty of the work and the danger of making a lower grade than usual are shared by all students in the class.

One year our school system paid for AP exams for any student making a B in the class (and reimbursing those who didn’t have the B, but made a 3 or better on the exam). We found that students with no financial obligation took the exams very casually. Some of those who took the classes during the fall term attended no review sessions, prepared very little, and scored a 1 on the exam in May. It would be a very serious mistake for the state to take on the huge investment of paying for exams which students could waste with impunity. If the QP is dependent on the exam score, having the public pay for the exam would be expected.

Most students know whether they have a good chance to pass the exam. They should be allowed to choose whether or not to take them. If all are required to take the exam, there will be more screening of those taking the classes. Those who would have trouble passing a college calculus course might be denied the chance to take it first in high school, which would have prepared them for success in college. The less talented student would be the loser again.

Some dual enrollment courses are offered by the community colleges. When students perceive these as less demanding than a high school class, they sometimes choose them as an easier way to get college credit than an AP course. Earning the college credit is al-ready a reward. Giving weighted grades should be done cautiously, with the degree of rigor and the demands of the course considered on an individual basis. AP and IB courses have very specific syllabi, which justify the weight.

Before the state made a uniform transcript, our school awarded a percentage of the student’s numerical grade as a weight. Five percent was the honors weight, and ten percent for AP classes. Some feel that was a fair approach. It gives everyone who passes the course some benefit, and it does reward those with higher grades proportionately.

This proposal should be considered very carefully. Its impact could be far-reaching.

FROM THE PRESIDENT’S DESK
Deborah Britt - Asheville High School 
 Asheville, North Carolina

My new job at Asheville High School is becoming very enjoyable. We have a paired AP Calculus/AP Physics course. We are on a 4 by 4 block except for these paired courses. AP History is paired with English, and there are others. I meet with one group of 15 students one day and a different 15 on the other day. So students have 2 days between classes for me and physics. I am surprised that this works quite well. If either of the teachers are away for meetings, the other teacher takes all 30 at one time. This also works well for test days and labs. I am hoping to spend more time with the physics teacher coordinating our topics and teaching some things together.

I have another AP Calculus class that just started January 4, and we have yet to take a derivative. I am very concerned about covering all material by AP Exam day of May 10. Please send any helpful hints to me or share them with others by writing to our newsletter.

I have recently learned a few things so I will pass them on to you for your thought time:

Slope fields are coming to the AB syllabus in 2003-04.

What is the domain of ? Had you ever thought about whether an exponential function should always have a positive base? It seems the TI-83 and TI-89 treats this problem differently.

The University of Illinois has a full selection of courses for post Calculus AB students. To view this information, visit  http://netmath.math.uiuc.edu 

Visit www.maa.org and click on the Journal of Online Mathematics and Applications (BRAND NEW)! There is a mathlet on Riemann sums.

Use a paper shredder to make strips for rectangular slices with area and volume.

Think about what the derivative of is.

The AP people are going to let me read out in Colorado again. So, I'll be grading lots of papers and learning some more mathematics. This will all happen about the third week in June.

Our next board meeting is scheduled for after the reading. If you have any ideas for us, please contact me or one of our board members. As always, it is a pleasure to work for you.

HISTORY OF THE ADVANCED PLACEMENT 
TESTING PROGRAM
Earl Mitchelle - The Asheville School 
 Asheville, North Carolina

The advanced placement testing program for calculus is fifty-seven years old. The first advanced placement examination given by the College Board was given in May of 1956. However, the program had been in place for two years prior to this test.

The Lawrenceville School (Lawrenceville, New Jersey), Phillips Exeter Academy (Exeter, New Hampshire), and Phillips Academy Andover (Andover, Massachusetts) joined together to write the first College Study of Admission with Advanced Standing test which was given in May of 1954. This testing program continued in 1955, before the College Entrance Examination Board (CEEB) and the Educational Testing Service (ETS) took over the program in 1956.

There was only one level of testing from 1956 through 1968. In 1969, separate AB and BC tests were given for the first time. Prior to 1983 calculators were not permitted on the examinations. Students were allowed to use scientific calculators on the 1983 and 1984 examinations, but the prohibition on calculators resumed in 1985 and ran through 1992. In 1993 and 1994 the use of scientific calculators was permitted again. In 1995, the use of graphing calculators was allowed for the first time. The “simple” graphing calculator has evolved into a very powerful tool with the addition of computer algebra systems (CAS) causing the Test Development Committee to become very innovative in designing questions that are calculator neutral. The design of the test changed for the 2000 examinations when the calculator active questions and non-calculator questions were used on both the multiple choice and free response sections.

What did the early test look like? Part I of the 1954 test had 30 short questions for which the student wrote the answers in a blanks next to the question; the time limit on this section was one hour. Part II consisted of 30 multiple choice questions with a one hour time limit. Part III had 5 free response questions with a one hour time limit.

The actual AP exams are copyrighted and cannot be posted on other sites without express permission from the College Board. NCAAPMT did obtain special permission to reprint the AP essays in our newsletter. If you would like back copies, please write Earl Mitchelle

STATISTICS FOR THE 2000 AP CALCULUS EXAMINATIONS

The Distribution Of Grades for All 2000 AP Calculus Candidates can be found on the internet under Grade Distributions at http://www.collegeboard.org/ap/calculus

This site also contains a lot of other helpful information including The Courses: Calculus AB and Calculus BC, The Exams: Calculus AB and Calculus BC, Teacher Resources, Student Resources, and Development Committee Members. This is a must-see site for all advanced placement calculus teachers.

The mean score, standard deviation, and percent of candidates getting a perfect score of nine (9) for each of the free response questions on the 2000 examinations are shown below. The questions common to both the AB and BC examinations are AB1/BC1, AB2/BC2, and AB5/BC5. Scores on these questions were used in the determination of the AB subscore for BC candidates. In addition, parts of question BC6 were used to determine the AB subscore for BC candidates.

AB Examination

   BC Examination

      The mean AB subscores for BC 6 are 2.25 out of 3 for parts a/b and 2.91 out of 6 for parts c/d.