NOTES FROM THE PRESIDENT'S DESK
Deborah G. Britt - Asheville High School - Asheville, NC

As your new President, I would like to take this opportunity to let you know that I want to hear from you.  Share your ideas with me and the other board members so that we can continue to serve you in the best way possible.  My address is:

Debbie Britt
326 Back Hollow Road
Mars Hill, NC 28754
Home Telephone & FAX: 828-689-5863
Email: dgb531@aol.com

I look forward to seeing you at the NCCTM meeting October 5th and 6th in Greensboro.  We have a 90 minute session on the program and will be talking about business items as well as the 2000 examinations and grading.  Please bring this newsletter with you to this session as we will have limited copies of the 2000 standards.  We were very fortunate this year to have Jeff Lucia who negotiated with ETS and College Board to obtain special permission for us to print the grading standards in our summer newsletter.  It is important that you hear from graders about the problems so that you can really appreciate the overall purpose of each question.  NCAAPMT will try to continue providing these insights to you at our meeting each fall.

As an AP grader for over 10 years and as a workshop consultant for College Board, I have learned a tremendous amount about the Test Development Committee's philosophy.  This really helps you to teach your students better mathematics - which is my primary focus.  We are lucky in that one of our own members is now on the Test Development Committee.  Ben Klein of Davidson College will be sharing some of his recent experiences at our session.  Hope to see you there.

We now have members in almost every state and four from foreign countries.  Our Web site (http://www.cctt.org/ncaapmt/) is up and running and will be improving.  It is really exciting how we have grown, although our number one focus is still on those members from beautiful North Carolina.  I live in the North Carolina mountains and have also taught in both the Central and Eastern regions of our state.  This year I have a new job as the AP Calculus teacher at Asheville High School and hope to be putting an AP Statistics course in place next year. I am looking forward to the challenge and to the help I will be getting from each of you to become a better teacher and in leading NCAAPMT.

Again, contact me if I can help or you want to share ideas.

A CONVERSATION WITH THE
AP CALCULUS TEST DEVELOPMENT COMMITTEE
Jeff Lucia - Providence Day School - Charlotte, North Carolina

On Wednesday, June 14 an open dialogue was held at the AP Calculus reading at Colorado State University between the members of the AP Calculus Test Development Committee (TDC) and nearly 600 AP Calculus readers and table leaders. The Committee members present were: Chairman Thomas Dick, Oregon State University, Corvallis, Oregon; David Bressoud, Macalester College, St. Paul, Minnesota; Mark Howell, Gonzaga College High School, Washington, DC; Benjamin Klein, Davidson College, Davidson, North Carolina; Martha Montgomery, Fremont Ross High School, Fremont, Ohio; Nancy Stephenson, Clements High School, Sugar Land, Texas; [not present] Stella Ashford, Southern University, Baton Rouge, Louisiana

Here are some of the highlights from that meeting. After the meeting the readers and the Committee adjourned to a social area for ice cream sundaes, and the discussion continued well on into the evening.

*     *     *

Reader: Will there be AP Exams in January any time soon?

TDC: Not likely. It's difficult enough to put together one exam per year without having to do two.

*     *     *

Reader: How are the exams put together?

TDC:: The first step is to put together a draft of the free response questions, then we revise them many times over.  We all submit questions, but when we get together (about four times a year) we pretty much beat up each other's questions.  We do invite anyone and everyone to submit suggestions and contributions, but by the time they appear on the exam, if they do, you might not recognize them.  After deciding on the free-response questions, the multiple choice questions are added to fill out the exam and cover as broad a range of topics as possible.  We pretty much have to have the 2001 exam finished by July of 2000.

*     *     *

Reader: What capabilities are allowed as far as calculators are concerned?

TDC: In the beginning, we said the calculators must be able to do four specific operations: sketch the graph of a function in an arbitrary viewing window, find the zeroes of a function, calculate the numerical value of the derivative of a function at a particular point, and calculate the value of a definite integral.  It was OK to have one or more of those operations accomplished by a program.  Later it was changed to be that the four operations had to be built in to the calculator.  There is no ceiling on the other capabilities, except that calculators with a QWERTY keyboard are not allowed.  This effectively eliminated the computer algebra system (CAS) calculators.  But now companies like Texas Instruments and Hewlett-Packard have CAS calculators without a QWERTY keyboard, so the job is much more difficult than before to put together an exam which is fair to the user of any of the allowable calculators.  Our job would be much easier if CAS calculators were required.  That's why it was necessary this year for the change to three free-response calculator-active questions followed by three non-calculator questions.  And in the calculator-active section, one of the  questions was really a non-calculator question (AB3 and BC3).  In the future, when and if CAS calculators are required, I expect that the Committee will embrace them and write exam questions which reflect their capabilities.

*     *     *

Reader: What are the biggest challenges you face as a committee?

TDC: There are really four. One is equity of the exam as it relates to calculator use.  Another is credibility to colleges.  Calculus courses differ widely from college to college as far as content and the extent to which technology is used.  We want our exam to be representative to as many colleges as possible, with students' success reflecting how well they might have done in just about any of the college courses nationwide.  A third challenge is teacher training as technological and curriculum changes have placed extraordinary demands on teachers.  And finally, we face the challenge of technology-active assessment.  Questions on the AP Exam must make essential and intelligent use of technology (AB1/BC1), rather than use it in a gratuitous way.

*     *     *

Reader: Why was there not a question this year which used tabular data similar to what we have seen on recent exams and for which there has been a big push in the revised syllabus?

TDC: Keep preparing your kids for them!  We just can't ask every type of question every year; actually, this year's AB2/BC2 was such a question in its original form.  But, as we said before, the questions take on many forms before they actually make it onto the exam.

*     *     *

Reader: The first three free-response questions (calculator-active section) seemed relatively easy and the last three (non-calculator section) seemed much harder.  I'm not sure the equal split in time (45 minutes each section) was really fair to the kids.  Couldn't you change the instructions to allow students to turn in their calculators when they are finished with the first three and let them get started on the other questions right away?

TDC: It would be pretty hard to implement a proctoring system which would allow that.  But this is something for which we need to look at for suggestions.

*     *     *

Reader: Will there ever be a student-produced answer grid-in section in place of some of the multiple choice questions as there are on the SAT?

TDC: We've never discussed that.  I suppose it's possible.

*     *     *

Reader: This year's questions were "recall and do" rather than emphasizing reasoning.  The exam looked more like the one in 1990 than 1997.

TDC: I'm not sure I agree with that, but I will say that we are still working on ways to fully embrace and incorporate the new syllabus into the exam.  We do try to spread the difficulty throughout each of the individual questions rather than increasing the difficulty from question 1 through question 6.  We hope that students will attempt to do every part of every question, or at least be able to get started on every question.  Perhaps AB1/BC1 is a little easier than the other questions.  It's not a bad thing for students to be able to get into the exam at the beginning.  Also, since 1995 roughly one third of the points have been calculator active.  They are now a little more compartmentalized, however, because of the new format.

*     *     *

Reader: Kids are losing a lot of points in successive parts of a question because they didn't get the correct answer to the first part.  Is it possible to ask a question in part (a), for instance, and then give the student an unrelated value with which to start part (b)?

TDC: The Committee somewhat likes the parts to be related, and we try to look at questions from a reader's perspective as to how much credit a student might lose for missing something in the early stages of a problem.

*     *     *

Reader: I was disappointed that the only mention of trigonometry on the entire free response section came in AB1/BC1, the function y = 1 - cos (x).

TDC:  We can't ask everything.

*     *     *

Reader: This year's questions seem too predictable, and the applications were unrealistic.

TDC: We have to be careful not to use questions which would favor a student who took AP Biology, for instance, over one who took AP Chemistry or AP Physics, or did not take an AP science course at all.  We need to keep the exam accessible to everyone, students and teachers alike.

*     *     *

Reader: Why couldn't the questions have been printed in the exam booklet? Students lost a lot of points because of transference errors.

TDC: This was tied to the split format (calculator vs. non-calculator).  We didn't want students to be able to look ahead to the last three problems
before putting away their calculators.  Other AP subjects have done this successfully for years, with all written answers in blue books.

*     *     *

Reader:  AB1/BC1 had three points for calculator computations.  Couldn't we return to the "Set up but do not integrate" type of questions?

TDC:  We may try to avoid heavy emphasis on calculator use in the future, but sometimes we want the student to do the calculation because we follow that with a question asking "What is the meaning of the answer?"  Also, telling the student to "Set up an integral expression" tells the student that the answer is an integral.  We would like them to know to use an integral.

*     *     *

Reader: Why are decimal approximations required to three places to the right of the decimal point?  This is not particularly appropriate for some applications.

TDC: Some standard has to be set, and that's what we have chosen.  That could be over-ridden in a certain problem, however, like the cola problem from a few years ago.  We are NOT going to significant figures!  That's an important concept, but not within the framework of our exam.  This year students lost points for having a correct exact form of an answer followed by a rounding error.  We will look at how we might change that for the future.

*     *     *

Reader: Is the exam, over the last five years or so, easier, harder or just different?

TDC: We have always used equating questions, usually from the multiple choice section, to compare and set the scale appropriately from year to year (these are questions which appear in identical form on consecutive exams). Generally, the students seem to do just about as well now as then.  Maybe it's because the exam is getting predictable or maybe it's that the students are getting better.  Probably it's a little of both.  One thing is for sure though, the teachers are better prepared to teach calculus in a technological environment than they were five or ten years ago.

*     *     *

Reader: Will there ever be questions on the exam for which a "shell" of preliminary information is given to students and teachers in advance of the exam, sort of like the case studies given in Computer Science?

*     *     *

TDC: We have not had any discussions about that yet.  It's an intriguing concept, though.  I'm glad you mentioned it.

MEAN SCORES AND ADJUSTED MEAN SCORES
FOR THE 2000 EXAMINATIONS

Each question on the free-response section of the examination is worth nine (9) points.  If a student does no work of a mathematical nature on a question, the student is given a - (dash) for this question on the grading form.  If a student does some work of a mathematical nature on a question, but this work earns no points, the student is given a 0 (zero) for this question  on the grading form.  The following table shows the mean score on each free-response question on both the AB and BC examinations including -'s and 0's and the adjusted mean score on each question based on grades of at least a one (1) point on the question.

 The total mean score on the
AB free-response questions: 24.50 out of 54
BC free-response questions: 33.53 out of 54

GENERAL NOTES FROM THE 2000 AP CALCULUS READING
Jeff Lucia - Providence Day School - Charlotte, North Carolina

*   There were about 170,000 exams total, 136,000 AB's and 34,000 BC's.

*   There is a decided shortage of college readers, hence some were invited back for a seventh consecutive year.  The number of available high school readers, however, is very large, and the new high school readers selected this year had originally applied in the years 1993-1995.

*   Any work on other paper or the green or blue exam inserts was not graded or even looked at. Only work in the exam booklet was considered.  There was some trouble with     students doing work on the wrong pages of the booklet or copying incorrectly from the green or blue sheets.

*   There was considerable debate over the correct use of parentheses and other mathematical notations, such as including dx in an integral, mixed use of labels in an integral, such as using t as both the independent variable in an integrand and the upper limit of the same integral (AB4), or attempting to remedy a negative answer for a volume by
continuing a  chain of equality which ultimately states "negative equals positive" (AB1/BC1 or AB4). The grading standards are becoming less forgiving about these types of errors  and their repeated occurrence could cost students several points throughout an entire exam.

*   There was also a lot of trouble with misrepresentation of decimal approximations.  The standards were relaxed a little this year to penalize only one point in a particular problem for presenting an answer to less than three decimal places, unless there were more different types of errors.  But no assumptions should be made as to how these will be viewed in future years.

*   A related error involved rounding in the middle of a problem that resulted in a final answer  which was not accurate.  Students could lose a point every time this occurred in a problem (AB1/BC1).

*   A few students lost points in AB1/BC1 for having their calculators in degree mode when graphing the function y = 1 - cos (x).

*   A related decimal problem which was unfortunate occurred when students had a correct answer in exact form followed by incorrect rounding, such as in AB2/BC 2, part (a): *.

*   The standards continue to stress that any type of two-dimensional graph or line graph sign test MUST be labeled.  Many justification points were lost over this (AB3, AB4). Verbal justifications must include specific reference to the correct function or derivative, not   just "the graph" or "it," and there must be a connection made with the behavior of a function to that of its derivatives, such as " f(x) increases on (2,3) because f '(x) is positive on that interval."

*   Students are reminded that no crossed out work will be graded. If there are several attempts to do a problem, students should highlight their answers to declare to the reader which one to read.  Leaving that decision to the reader can cost a student points.

*   Readers often go to great lengths to read with a student in successive parts of a problem which depend on an earlier answer.  However, justification points are at risk because a student cannot justify that an incorrect answer is correct (AB4, part (d): "the absolute maximum occurs at t = 65 because..."  That student's derivative does not change signs at  t = 65.

*   There was concern about 3 points being awarded for calculator answers to AB1/BC1.

*   Readers suggested that the tick marks be omitted from given graphs, because in  AB1/BC1 many students mistakenly assumed the two graphs intersected at x = 1 and in AB2/BC2 students incorrectly estimated a y-value from the graph.

*   Points are not awarded for a student's reciting formulas as part of a solution: "Take the derivative, set it equal to zero and solve," for example.  Points may be awarded when the student does that with the specific function in a problem.  Because many labels like f, s, v, or a appear frequently as formulas, future exams may use different labels for specific functions in a given problem.

*   The presence of a non-calculator section of free-response questions enabled the exam to cover more topics in what many readers felt was a better or more realistic way.

*   Students should probably try to at least minimally justify all answers, whether or not the question specifically asks for a method or justification.

*   There is a continued desire on the part of the College Board for teachers to teach a thorough course in calculus and not simply to teach for the AP examination.