Students’ scores on AP calculus examinations are widely accepted by college and university mathematics departments for placement and/or credit. The amount of credit and final placement in advanced courses are dependent on which AP examination a student takes (AB or BC), the score that he/she earns, and the college or university where the student seeks credit or advanced placement. Research conducted by the College Board and others (see for example, University of California at Los Angeles 2000) supports accepting scores of 4 or 5 for credit and/or placement. In one study, researchers administered the same calculus problems to AP calculus students who earned a score of 4 or 5 on the AP examination and students who completed a college calculus course. This study revealed that AP students who scored a 4 or 5 on the AP examination performed on these questions as well as or better than most of their college counterparts who received honor grades in the comparable college courses (College Board, 1999).

The panel agrees with departmental policies that accept scores of 4 or 5 on an AP calculus examination as evidence that students have mastered knowledge and skills equivalent to those presented in an introductory college calculus course. Through legislative mandate, however, some publicly funded colleges are required to award credit or placement for scores of 3, and the panel is not convinced that such scores reflect sufficient mastery of the subject to warrant credit or placement. Students who earn a score of 3 or lower on an AP examination or 5 on an IB examination can present problems for colleges in terms of placement in introductory courses. The panel suggests that credit or placement should be awarded to students who earn a score of 3 on the AP examination on an individual basis and should be supported by additional evidence beyond examination performance.^{29}

Scores earned on IB mathematics examinations are treated similarly to scores earned on AP calculus examinations. It should be noted, however, that the different levels at which IB courses and examinations can be taken and the variety of optional topics that are included in different classrooms add further considerations for awarding of credit and/or placement, as it is more difficult for a university mathematics department to know exactly what an IB student has learned.

The calculus portion of IB Mathematics HL, including the Approximation and Analysis option, is generally considered to be at least the equivalent of AP Calculus BC. The calculus portion of Mathematical Methods SL by itself is less than a full AP Calculus AB course. However, the majority of Mathematical Methods SL teachers in the United States teach a full calculus class that is equivalent to an AB Calculus course, even though this additional material is not required for the IB examination. Students who earn scores of 6 or 7 are usually regarded as having passed the equivalent of a college-level course. The exact course for which credit and further placement are awarded depends on the IB course and examination for which the student

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools - Report of the Content Panel for Mathematics
ACCEPTANCE OF STUDENTS’ AP AND IB TEST RESULTS BY COLLEGES AND UNIVERSITIES
Students’ scores on AP calculus examinations are widely accepted by college and university mathematics departments for placement and/or credit. The amount of credit and final placement in advanced courses are dependent on which AP examination a student takes (AB or BC), the score that he/she earns, and the college or university where the student seeks credit or advanced placement. Research conducted by the College Board and others (see for example, University of California at Los Angeles 2000) supports accepting scores of 4 or 5 for credit and/or placement. In one study, researchers administered the same calculus problems to AP calculus students who earned a score of 4 or 5 on the AP examination and students who completed a college calculus course. This study revealed that AP students who scored a 4 or 5 on the AP examination performed on these questions as well as or better than most of their college counterparts who received honor grades in the comparable college courses (College Board, 1999).
The panel agrees with departmental policies that accept scores of 4 or 5 on an AP calculus examination as evidence that students have mastered knowledge and skills equivalent to those presented in an introductory college calculus course. Through legislative mandate, however, some publicly funded colleges are required to award credit or placement for scores of 3, and the panel is not convinced that such scores reflect sufficient mastery of the subject to warrant credit or placement. Students who earn a score of 3 or lower on an AP examination or 5 on an IB examination can present problems for colleges in terms of placement in introductory courses. The panel suggests that credit or placement should be awarded to students who earn a score of 3 on the AP examination on an individual basis and should be supported by additional evidence beyond examination performance.29
Scores earned on IB mathematics examinations are treated similarly to scores earned on AP calculus examinations. It should be noted, however, that the different levels at which IB courses and examinations can be taken and the variety of optional topics that are included in different classrooms add further considerations for awarding of credit and/or placement, as it is more difficult for a university mathematics department to know exactly what an IB student has learned.
The calculus portion of IB Mathematics HL, including the Approximation and Analysis option, is generally considered to be at least the equivalent of AP Calculus BC. The calculus portion of Mathematical Methods SL by itself is less than a full AP Calculus AB course. However, the majority of Mathematical Methods SL teachers in the United States teach a full calculus class that is equivalent to an AB Calculus course, even though this additional material is not required for the IB examination. Students who earn scores of 6 or 7 are usually regarded as having passed the equivalent of a college-level course. The exact course for which credit and further placement are awarded depends on the IB course and examination for which the student
29
The College Board currently is conducting research on the validity of awarding credit or placement for a score of 3. The panel did not have access to any data from that study. Its results should be published within a year.

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools - Report of the Content Panel for Mathematics
submits a score. Since many fewer students take IB than AP, acceptance of the IB score is often based on analogy with AP, rather than on an evaluation of the particular strengths of the IB course. There is likely to be less topic-by-topic correspondence between IB courses and college courses than is the case for AP because of differences in the ways in which the IB and AP courses have been developed. However, research and experience appear to indicate that students who have been successful in the IB curriculum are as well prepared for further mathematics as those who have taken AP.30
Neither advanced placement nor acceleration is a program goal of the IB program. The IBO does not actively encourage colleges and universities to accept IB examination results for college credit and/or advanced placement in upper-level mathematics courses. Rather, the acceptance of these credits resulted through analogy to AP (conference notes, speech by Paul Campbell, Director of Operations, IBNA). Acceptance occurred slowly as parents, teachers, and students, on an individual basis at first, demonstrated to college registrars and mathematics departments that IB courses were equivalent to AP courses in rigor and content. It was this comparison that led to the acceptance of IB examination scores. Thus, understanding the use of IB examination scores for purposes of college credit and placement also requires understanding how AP gained widespread acceptance.
The College Board has done an admirable job in helping to establish and maintain communication between college and high school mathematics faculties so they can stay abreast of each other’s interests and concerns. Members of the AP test development committee, which is responsible for the AP calculus program, are drawn from both college and high school faculty. Every year a group of several hundred college and high school faculty meet for a week or more in early June to grade the free-response questions of that year’s AP examination. This provides an excellent opportunity for informal exchanges on mathematics and teaching and allows the College Board to monitor the reactions of college and high school faculty, as well as student responses to questions on the examination.31
Finding: Most college and university placement and credit practices that are based on student performance on AP calculus or IB mathematics examinations are reasonable.
UNINTENDED USES OF AP AND IB ENROLLMENT FIGURES AND EXAMINATION RESULTS
The media and others sometimes use AP and IB scores inappropriately as measures of quality for factors that are not related to the intended purposes of the courses or examinations. For example, some members of the media have proposed using the number of AP and IB courses
30
We compare the preparation level of good IB students with that of good AP students because college and university faculty have had much more extensive experience with AP calculus than with IB mathematics. The panel is not implying that AP is the standard to be met.
31
The College Board views its experiences with the revision of the AP calculus program as a potential model for ways in which to approach the redesign of other AP courses (College Board, 2001). A detailed historical perspective on the communications among the College Board, mathematicians, teachers, and professional mathematics organizations that led to the creation of the new AP calculus courses was written by Dan Kennedy of the Baylor School, and can be found on the College Board’s Web site: www.collegeboard.com.

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools - Report of the Content Panel for Mathematics
offered in schools as a tool to rate and rank the schools against each other, possibly leading some schools to offer more AP or IB courses than they are able to support adequately with existing resources. Some schools use the results of AP or IB examinations to evaluate teachers, the consequence of which may be to discourage potentially low-scoring students from taking the courses or the examinations. It is not uncommon for selective colleges to view the existence of AP or IB courses on an applicant’s transcript as an important part of their evaluation of the student’s intellectual and academic motivations. Without any measure of the quality of the student’s achievement in such courses,32 however, this emphasis on the number of AP or IB courses on a transcript leads many students to enroll in the courses without a commitment to mastering the material. AP and IB enrollments are increasingly used as a policy lever, as a method of raising the standards and expectations for all students. This leads to negative consequences for students for whom advanced courses in general, and AP and IB mathematics in particular, may not be appropriate.
Finding: Data on the number of AP and IB courses offered by schools and the results of the examinations are sometimes used in ways for which they were not intended, thus creating situations that can be detrimental to student learning.
32
Results of AP examinations taken in the senior year are not available until long after college acceptances have been sent out.

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools - Report of the Content Panel for Mathematics
6
Recommendations
PREPARATION OF STUDENTS
All calculus taught in high school should be at the college level.
All students who enroll in AP calculus should have had at least 4 years of college preparatory mathematics prior to AP calculus. The structure of IB mathematics courses is different, and this recommendation is not as applicable to them.
Strategies must be developed to ensure that students who enroll in calculus have an adequate background in algebra and trigonometry for subsequent work in mathematics and science.
PROFESSIONAL DEVELOPMENT
The professional development opportunities available to AP and IB teachers should be improved and expanded.
The College Board and the IBO should ensure that adequate professional development opportunities are available to all AP and IB teachers.
Schools that choose to offer AP and IB programs must find ways to encourage all teachers to take part in professional development, perhaps by providing time during the school day rather than on nights and weekends.
The College Board should consider developing procedures to certify AP calculus teachers.
ASSESSMENT
The AP and IB examinations should vary more from year to year. If teachers expect that major ideas will be assessed rather than specific problem types, it is likely that instruction will encourage the development of students’ critical thinking and problem-

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solving abilities. The AP and IB examinations must strike a balance between judging students’ conceptual understanding by asking unfamiliar probing questions, and alarming teachers and students with a strange and unfamiliar test.
Both the AP and IB programs should maintain and increase their focus on conceptual understanding in their assessments.
To assess computational and procedural knowledge, students should be asked questions that demonstrate their ability to use these procedures in solving more complex problems.
All students enrolled in AP and IB courses ordinarily should take the relevant external examinations as part of the course requirements.33
AP and IB examinations should contain more emphasis on realistic applications, including those in which student have to set up the mathematical model.
33
The panel recognizes that policies must be developed to ensure that students without the financial means to take the tests are not penalized by this requirement. The panel notes that the College Board is investigating ways to increase the numbers of students who take the AP examination at the conclusion of an AP course. For more information, see the report of the Commission on the Future of the Advanced Placement Program, which is available at www.collegeboard.com/.

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools - Report of the Content Panel for Mathematics
Appendix A
Charge to the Mathematics Panel from the Parent Committee
Charge to the Parent Committee and Content Panels: The charge to the Committee is to consider the effectiveness of, and potential improvements to, programs for advanced study of mathematics and science in American high schools. In response to the charge, the committee will consider the two most widely recognized programs for advanced study: the Advanced Placement (AP) and the International Baccalaureate (IB) programs. In addition, the committee will identify and examine other appropriate curricular and instructional alternatives to IB and AP. Emphasis will be placed on the mathematics, physics, chemistry, and biology programs of study.
Charge to Content Panels: The content panels are asked to evaluate the AP and IB curricular, instructional, and assessment materials for their specific disciplines.
Below is a list of questions that the content panels will use to examine the curriculum, laboratory experiences, and student assessments for their specific subject areas. The content panels will use these questions to issue a report to the Committee about the effectiveness of the AP and IB programs for educating able high school students in their respective disciplines. In answering these questions, the content panels should keep in mind the Committee’s charge and study questions.
The panels should focus on the following specific issues in advising the Committee:
I. CURRICULAR AND CONCEPTUAL FRAMEWORKS FOR LEARNING
Research on cognition suggests that learning and understanding are facilitated when students: (1) have a strong foundation of background knowledge, (2) are taught and understand facts and ideas in the context of a conceptual framework, and (3) learn how to organize information to facilitate retrieval and application in new contexts (see, e.g., Bramsford et al., 1999; handouts in packet).
To what degree do the AP and IB programs incorporate current knowledge about cognition and learning in mathematics and science in their curricula, instructions, and assessments?