NOTES FROM THE 1996 ANNUAL MEMBERSHIP MEETING
Jane Barnett - President

The annual meeting in Greensboro, October 4, was a successful event thanks to several fine presenters who contributed. There were approximately 40 people in attendance. The current membership went over the 170 mark with the addition of some new members.

Earl Mitchelle, Executive Secretary, read the proposed amendments to the constitution which had been mailed to the membership priol to this meeting. The amendments were approved by a unanimous vote of the members present. Briefly, these amendments provide for six board members, an addition of two board members, in addition to the five officers. They also mandate that at least two board members, other than the Executive Secretary and Treasurer, represent each of the three regions of the North Carolina Council of Teachers of Mathematics(NCCTM). Three representatives will be appointed each year for two-year terms. A representative may serve a maximum of three two-year terms unless the representative is elected an officer.

Fr9m the Eastern Region we welcomed Frank Vrablic to the Board as a Representative. Mr. Vrablic teaches at Manteo High School where he is the coach of the school's outstanding mathematics team. One of the new positions on the Board will be filled by Debbie Britt who teaches at Mitchell High School in the Western Region. Ms. Britt is an AP grader and presents College Board workshops, The second of the new positions will be filled by Sue Sams of Providence High School in Charlotte in the Western Region. She is also an AP grader. I am pleased to announce their joining the Board.

Jean Taylor of Wilmington who is retiring from the Board was recognized for her many contributions to the teaching of mathematics. She is a master teacher, AP grader, presenter at NCCTM conferences, and contributor to the NCA2PMT Newsletter. Ms. Taylor long recognized the need for a network of advanced placement mathematics teachers in North Carolina and was happy to serve as a charter board member of the organization when it was started by Charlie Bodine.

Melba Tripp was also recognized for her leadership for the past four years as President Elect and President. Ms. Tripp will continue to provide valuable advice and assistance as Past President for the next two years. She is currently teaching at East Carolina University.

Thanks were extended to the Professional Engineers of North Carolina for a $300 grant which covered the cost of printing and mailing one issue of the newsletter. The generous donation was in keeping with the tradition of the organization which also supports the Mathcount program which encourages younger students in the state.

Steve Davis, Earl Mitchelle, and Sharon Walker, AP graders, reviewed the grading stand ards for several free-response questions on the 1996 AB and BC examinations. They also discussed some of the trends and issues related to the constantly changing examinations. Debbie Britt discussed the new course description which will become effective for the 1998 examinations. She also presented several calculator-active problems. The various perspectives of these fine members of NCA2PMT make these sessions not-to-be-missed opportunities.

SCA2PMT and NCA2PMT plan another joint meeting at the NCCTM annual conference to be held in Charlotte in the fall of 1997.

As you enjoy the rest of this newsletter, consider what information or natenal you might contribute to future issues. We are all indebted to the editor, Earl Mitchelle, for the tenacious way he seeks out items of interest for the newsletter. This valuable resource is surely one of the reasons our membership includes educators from a dozen states other than North Carolina.

MEMORY LANE

How has the advanced placement examination in mathematics evolved over the years? The advanced placement examination program was born in the early 1950's. For a number of years, there was only one examination for mathematics. In 1969, the AB and BC examinations were first given.

Debbie Britt, a NCA2PMT Board member and College Board workshop instructor, has put together a package of AP calculus examinations which have been given over the past 40 years. She has included the 1957 and 1967 from the pre-ABIBC era and the 1977 and 1987 AB examinations. Has the degree of difficulty of the AP calculus examinations decreased with the passing of time or has the emphasis just changed? How much different will the 1997 examinations be relative to the 1957 examination?

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.

UNITS AND AP EXAMINATIONS

Questions On both the AB and BC advanced placement examinations are requiring students to give the proper units for certain answers. The grading standards are frequently set up to require that the student get both the numerical value and the units for this value correct in order to earn the answer point. A correct numerical value with no units or incorrect units does not earn the answer point. It is necessary for students to know how to determine the correct units when asked to do so.

When a graph is drawn, the values of the independent variable are recorded on the horizontal axis while values of the dependent variable are recorded on the vertical axis. To find the slope of a straight line, two points on the line are selected and the difference of the dependent variable values for these two points is divided by the difference of the independent variable values for these two points, in the same order.

The units of the slope will be the units of the variable on the verticalidependent variable axis divided by the units of the variable on the horizontal/independent variable axis. If the position of a particle executing rectilinear motion is plotted on the vertical axis with units of centimeters and the time is plotted on the horizontal axis with units of seconds, the units of the slope will be centimeters divided by seconds or centimeters per second which are the units for the velocity.

If the area bounded by a function of some independent variable and the horizontall independent variable axis between two value of the independent variable is calculated, the units of the answer will be the units of the variable on the independent axis multiplied by the units of the variable on the dependent axis.

Returning to the situation of a particle executing rectilinear motion, if the velocity of the particle is plotted as a function of time, the area bounded by the function of the velocity and the horizontal axis between two values of time will have units which equal the units of the variable on the horizontal axis, seconds in this case, multiplied by the units of the variable on the vertical axis, centimeters per second in this case. Seconds multiplied by centimeters per second yields centimeters, and the area under the curve found between the two values of time selected as the left hand and right hand boundaries represents the distance traveled by the particle during that time period.

The slope of the velocity versus time graph for a particle executing rectilinear motion also has meaning. Centimeters per second divided by seconds yields centimeters per second per second or centimeters per second² which is the acceleration of the particle.

There are times when the slope of the graph of a function and the area under the function will not be useful or meaningful values. The area under the position function has units found by multiplying centimeters by seconds yielding centimeter-seconds which has no meaning.

The units for the value of the derivative of a function are equal to the units of the variable on the vertical axis divided by the units of the variable on the horizontal axis. The units of the definite integral of a function equal the units of the variable on the vertical axis multiplied by the units of the variable on the horizontal axis. Sometimes these values with their units have useful interpretations while at other times they do not.

The determination of units for a derivative or definite integral can be looked at from a slightly different direction. For the function y = f(x), one of the notations for the derivative of f(x) is dy/dx. This suggests that the units for the derivative can be determined by dividing the units for y by the units for x. In the expression for the integral of f(x), f(x) and dx appear as f(x) dx suggesting that the units for the definite integral can be found by multiplying the units of f(x) by the units of x.

The 1996 free-response sections of both the AB and BC examinations contained questions about units. These questions were AB3/BC3, AB5, and BC5.

ANNUAL MEETING OF THE BOARD OF DIRECTORS
Earl Mitchelle

The Board of Directors of the North Carolina Association of Advanced Placement Mathematics Teachers held its annual meeting on August 17, 1996, at Western Guilford High School in Greensboro, North Carolina.

The annual Treasurer's report showed a gain of $209.00 for the fiscal year which endmg June 30, 1996. At this time there were 145 active members and 64 inactive members of NCA2PMT.

President Melba R. Tripp appointed Charles H. Bodine, Bernice H. Kenan, and Frank Vrablic to two-year terms as membership representatives for 1996-1998. Carolyn M. Walmsley was elected President Elect by the Board. Members of the Board of Directors are as follows:

Jane R. Barnett, Laurinburg, NC - President
Carolyn M. Walmsley, Greensboro, NC - President Elect
Melba R. Tripp, Greenville, NC - Past President
W. Earl Mitchelle, Asheville, NC - Executive Secretary
Geoffrey A. Lucia, Charlotte, NC - Treasurer
Margaret Wirth; Greenville, NC - Representative
Bernice H. Kenan, Greensboro, NC - Representative
Charles H. Bodine, Charlotte, NC - Representative
Frank Vrablic, Manteo, NC - Representative
Deborah Britt, Mars Hill, NC, Representative*
Sue Sams, Charlotte, NC - Representative*

* Ms. Brift and Ms. Sams were appointed to the Board as Representatives after the constitutional amendments changing the composition of the Board were approved at the Annual Meeting on October 4, 1996.

The Board voted to recommend to the membership that the number of representatives on the Board be increased from four to six. The Executive Secretary and Treasurer are permanent members of the Board, and, at the present time, are both from the western region. The addition of two representatives to the Board will insure that each of the three regions of the state will be represented by individuals who periodically change. Notice of these proposed changes to the Constitution were sent to the membership on September 1,1996. The membership voted on the proposed changes at the NCA2PMT session at the NCCTM conference in Greensboro, North Carolina, on October 4, 1996. The result of this vote is reported in this issue of the newsletter.

Jean F. Taylor of Wilmington, NC, retired from the Board after serving as a membership representative from the eastern region since the founding of NCA2PMT.

Plans for NCA2PMT's participation in the regional meetings of NCCTM in the spring of 1997 and the NCCTM state conference in the fall of 1997 were discussed.

The next meeting of the Board is tentatively scheduled for June of 1997 in Greensboro, NC.

MISCELLANEOUS DATA AND TRIVIA

• Nearly 1 in 5 students entering four-year colleges is eligible for AP credit.
• Approximately 52% of high schools offer AP courses.
• 46% of the 1995 graduates had 2 or more AP grades on their high school transcripts. English, history, and calculus are the most popular courses followed by biology, Spanish, and chemistry.
• More females than males take AP examinations, and their numbers are rising faster than those of males.
• Minority participation in the AP examinations has increased from 19% of test takers to 29% in the last 10 years.
• AP students' averages on SAT's recentered scale were 627 Verbal and 624 Math. 81 % of AP students were in the top fifth of their high school class.
• SAT:l Reasoning Test scores for AP students were 100 points above the national average in reading and mathematics

• 61% of the candidate grades were 3 or better on the 1996 AB Calculus Examination.
• 80% of the candidate grades were 3 or better on the 1996 BC Calculus Examination.
• In North Carolina in 1996 the number of schools, candidates, and exams decreased by 2%, 1% , and 1%, respectively, from 1995. State funding for the AP program was lost. However, there was a 7% increase in the candidates who got a grade of 3 or better. In Georgia where state funding was also lost, there was no change in the number of schools, but the number of candidates and exams decreased by 24% and 25%, respectively, from 1995 to 1996 while grades from 3 to 5 decreased by 5%.

BLOCK SCHEDULING
Earl Mitchelle

A letter from Thomas A. Struble of Unionville High School in Kennett Square, PA, appeared in the October 1996 edition of The Mathematics Teacher and summarized his school's reasons for not adopting block scheduling for mathematics. After a two-year study the faculty voted against block scheduling by a two-to-one margin.

The reasons given for not adopting block scheduling are

1. Claims for the benefits of block scheduling are unsubstantiated and anecdotal;

2. Objective studies in Canada showed that test scores declined where block scheduling was used.

3. As much a 35%-40% less material is covered in a block schedule than in a traditional schedule. A 35%-40% annual coverage shortfall compounded over four years of high school translates into a very weak education in mathematics.

The Newsletter contacted Struble and requested more information about his department's research. Some of the key points are

1. The periods in a block schedule are too long for mathematics.

2. The idea that less material would be covered in a block schedule but would be covered better is not "less is more" but is "less is less."

3. Claims that discipline problems and drop out rates would decline and attendance and grades would improve in a block schedule could not be substantiated because of insufficient evidence.

4. Some advocates of block scheduling claim that the work load and homework for students is reduced. "Less is less?"

Diane Webb of Belton, Missouri, reported that the failure rate for first-year algebra students who completed a full-year course in 18 weeks was so high, that her school abandoned the block schedule.

Larry Romary of Monroeville, Indiana, stated that in his school, he finished 12 chapters in a first-year algebra textbook using the traditional 50-minute schedule while other teachers finished only 8 chapters in the same textbook in a block schedule. He feels that the shortfall comes because it is not possible to teach two sections in an 85-minute period. He also added that the teacher's manual for his textbook warns against trying to use a block schedule.

Another teacher notes that a student who takes a full-year course in the first semester in a block schedule, takes no mathematics in the second semester, and resumes taking mathematics in the first semester of the following year frequently encounters a lot of difficulty. This teacher also noted that class size tends to increase when a block schedule is implemented because the same number of students are divided into four classes instead of six classes.

A student in Wyoming expressed her dislike for block scheduling because she had lost time for band and other artistic classes. She added that one or two days of illness can cause a student to miss up to four days of work instead of the usual two days of work. The block schedule makes her feel "frustrated, rushed, and exasperated a great deal of the time."

In British Columbia, a study of the standardized test results was done for 20,000 students taking Chemistry 12. The examination grades for students in this course for time spent studying the course are full year- 70%, semester - 64%, and quarter - 57%. Similar exam results achieved for Physics 12, Math 12, and English 12.

David Boldt, a writer for the Philadelphia Inquirer, writes that College Board "reported that students in schools that use block scheduling who take advanced placement tests tend to do poorly in most subjects when compared with students from high schools that do not use block scheduling. The exception is English where there is no significant difference." Boldt went on to say that the difference is equivalent to a 50-point difference in terms of SAT scores. He adds that teachers of advanced placement courses "overwhelmingly oppose" block scheduling.

Boldt goes on to say that there is no evidence that block scheduling improves student achievement, but there is a growing body of evidence that it can do harm.

Alan Kors, a professor at the University of Pennsylvania, states about block scheduling, "This is one of the most extraordinary frauds I have ever seen, and I have seen a lot of them." He says that courses taught in a block schedule that require constant repetition to learn such as mathematics, science, and foreign languages can end in "disaster."

David Bateson, a professor of education at the University of British Columbia, found that 10th. graders in the province who take a full-year course do better than students who take a semester course in a block schedule.

The NCTM News Bulletin reported in its September 1996 edition that a study in North Carolina revealed that end-of-course test results in 1994 for students in block schedules in mathematics were lower than those for students in traditional courses.

Block scheduling seems to be popular with administrators but not with classroom teachers, particularly advanced placement teachers. When students do the course material for AP calculus in the fall semester, it does not take great insight to see that these students will not do well on the examination which is given in May unless they do review work during the second semester. At the grading of the examinations at Clemson University last summer, College Board asked graders about whether they would be available to do grading under some system in January or February. It appears that College Board is doing some thinking about this problem, but any changes in the testing schedule is probably years away if there are ever any. The cost of developing two examinations and grading them would certainly lead to higher costs for students.