1
Introduction

The National Research Council’s (NRC) Committee on Programs for Advanced Study of Mathematics and Science in American High Schools (parent committee) formed the mathematics panel to provide advice on the effectiveness of and potential improvements to programs for the advanced study of mathematics in U.S. high schools.2 In accordance with its charge (see Appendix A) the panel focused its work on the Advanced Placement (AP) and International Baccalaureate (IB) programs.

A member of the parent committee chaired the panel. This individual also served as the panel’s liaison to the committee and consolidated the panel’s findings and recommendations into this report. The five other panel members represent university and high school mathematics faculty with a wide range of teaching experiences and a diversity of perspectives on mathematics education. Biographical profiles of the panel members can be found at the end of this report.

In conducting this study, the panel examined two AP courses—Calculus AB and Calculus BC—and three of the four mathematics courses offered by the International Baccalaureate Organisation (IBO)—Mathematical Methods SL,3 Mathematics HL, and Further Mathematics SL.4 The panel’s analysis was informed primarily by a range of program materials, such as published mission statements, course outlines, teacher guides, sample syllabi, released examinations, and scoring rubrics provided by the College Board and the IBO; scholarly research related to the teaching and learning of mathematics; and the combined experience and expertise of the panel members (College Entrance Examination Board, 1987; 1989; 1992; 1993; 1994;1995; 1996a; 1996b; 1997a; 1997b; 1998a; 1998b; 1999a; 1999b; 1999c; 1999d; 1999e; 199f; International Baccalaureate Organisation, 1993; 1997; 1998a; 1998b; 1998c; 1998d; 1999a; 1999b; 1999c; 1999d; 1999e; 1999f; 1999g; 1999h; 1999i; 1999j; 1999k; 1999l; 1999m; Kennedy, 1997). The panel also considered two College Board studies addressing AP student achievement in college (Morgan and Ramist, 1999; Willingham and Morris, 1986) and several studies that examined the subsequent performance of IB students in college (Guild of IB

2  

The mathematics panel is one of four panels convened by the parent committee. Each panel represented one of the four disciplines that were the focus of the committee’s study: biology, chemistry, physics, and mathematics. The reports of the other panels and the report of the full committee are available at www.nap.edu/catalog/10129.html.

3  

It is the hours of contact time, not necessarily the level of course difficulty, that determines whether a course is designated as SL or HL.

4  

The panel did not review Mathematical Studies SL because it is a basic rather than an advanced course.



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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 1 Introduction The National Research Council’s (NRC) Committee on Programs for Advanced Study of Mathematics and Science in American High Schools (parent committee) formed the mathematics panel to provide advice on the effectiveness of and potential improvements to programs for the advanced study of mathematics in U.S. high schools.2 In accordance with its charge (see Appendix A) the panel focused its work on the Advanced Placement (AP) and International Baccalaureate (IB) programs. A member of the parent committee chaired the panel. This individual also served as the panel’s liaison to the committee and consolidated the panel’s findings and recommendations into this report. The five other panel members represent university and high school mathematics faculty with a wide range of teaching experiences and a diversity of perspectives on mathematics education. Biographical profiles of the panel members can be found at the end of this report. In conducting this study, the panel examined two AP courses—Calculus AB and Calculus BC—and three of the four mathematics courses offered by the International Baccalaureate Organisation (IBO)—Mathematical Methods SL,3 Mathematics HL, and Further Mathematics SL.4 The panel’s analysis was informed primarily by a range of program materials, such as published mission statements, course outlines, teacher guides, sample syllabi, released examinations, and scoring rubrics provided by the College Board and the IBO; scholarly research related to the teaching and learning of mathematics; and the combined experience and expertise of the panel members (College Entrance Examination Board, 1987; 1989; 1992; 1993; 1994;1995; 1996a; 1996b; 1997a; 1997b; 1998a; 1998b; 1999a; 1999b; 1999c; 1999d; 1999e; 199f; International Baccalaureate Organisation, 1993; 1997; 1998a; 1998b; 1998c; 1998d; 1999a; 1999b; 1999c; 1999d; 1999e; 1999f; 1999g; 1999h; 1999i; 1999j; 1999k; 1999l; 1999m; Kennedy, 1997). The panel also considered two College Board studies addressing AP student achievement in college (Morgan and Ramist, 1999; Willingham and Morris, 1986) and several studies that examined the subsequent performance of IB students in college (Guild of IB 2   The mathematics panel is one of four panels convened by the parent committee. Each panel represented one of the four disciplines that were the focus of the committee’s study: biology, chemistry, physics, and mathematics. The reports of the other panels and the report of the full committee are available at www.nap.edu/catalog/10129.html. 3   It is the hours of contact time, not necessarily the level of course difficulty, that determines whether a course is designated as SL or HL. 4   The panel did not review Mathematical Studies SL because it is a basic rather than an advanced course.

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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 Schools, 2000.5 The panel found surprisingly few other data on AP and IB on which to base its evaluation.6 For example, little is known, except anecdotally, about how either program is implemented in U.S. high schools, including the instructional strategies and resources used in individual classrooms, the structure of the syllabi in different schools, the quantity and quality of the facilities available, the preparation of teachers who teach the courses, and the ways in which students are prepared prior to enrolling in AP calculus or advanced IB mathematics courses. Information about the AP and IB assessments is also limited. The IBO currently conducts no systematic research addressing the validity,7 reliability,8 or comparability9 of its assessments across administrations (Pook, 2001). The senior examiners, not psychometricians, make determinations about the degree to which each administration is a valid and reliable measure of student achievement. The College Board, on the other hand, has gathered considerable data to demonstrate the reliability and comparability of student scores from one administration to the next and from one student to another. However, neither program has a strong program of validity research, and neither has gathered data to document that the test items on its examinations measure the skills and cognitive processes they purport to measure. To fully analyze or evaluate the AP and IB assessments, it is necessary to know, for example, that test items intended to measure problem solving do in fact tap those skills and do not just elicit memorized solutions or procedures. Further, little evidence is available for evaluating the long-term effects of the AP and IB programs. For instance, the panel could not find systematic data on how students who participate in AP and IB fare in college mathematics relative to other students. Nor could the panel find studies that examined the effects on postsecondary mathematics programs of the ever-increasing numbers of students who are entering college with credit or advanced standing in mathematics. While the College Board and a few colleges that receive IB students have conducted some isolated studies addressing how AP or IB students perform in college (see, for example, Morgan and Ramist, 1998), the inferences that can accurately be drawn from the findings of these studies are ambiguous (see Chapter 10 of the parent committee’s report). Because empirical evidence about the programs’ quality and effectiveness is lacking, the panel focused its analysis on what the programs say they do, using available program materials. 5   Downloaded from www.rvcschools.org in August 2000. 6   Although few data currently exist, the panel notes that both programs have circulated requests for proposals to conduct research on the ways in which their respective programs are implemented in schools and classrooms and the effects of these different implementations on student learning and achievement. No data from any of these studies were available at the time this report was prepared. 7   Validity addresses what a test is measuring and what meaning can be drawn from the test scores and the actions that follow. It should be clear that what is being validated is not the test itself, but each inference drawn from the test score for each specific use to which the test results are put. 8   Reliability generally refers to the stability of results. For example, the term denotes the likelihood that a particular student or group of students would earn the same score if they took the same test again or took a different form of the same test. Reliability also encompasses the consistency with which students perform on different questions or sections of a test that measure the same underlying concept, for example, energy transfer. 9   Comparability generally means that the same inferences can be supported accurately by test scores earned by different students, in different years, on different forms of the test. That is, a particular score, such as a 4 on an AP examination, represents the same level of achievement over time and across administrations.

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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 This report details the panel’s findings about the programs based on its analysis of these materials, its advice to the committee regarding the major issues under its charge, and the types of evidence that are needed to enable more complete appraisal of the AP and IB mathematics programs—evidence that is urgently needed. Before proceeding to report its findings and recommendations, the panel wishes to commend the College Board and the IBO for the ways in which they have responded to changes in the teaching of mathematics: AP by changing its syllabus and introducing technology; IB by introducing portfolio projects and technology. Both organizations have undertaken these changes in a balanced, reasoned way, with input from a wide range of communities. Considering that they are large organizations affecting thousands of students, teachers, universities, and colleges, both AP and IB have made significant and forward-looking changes remarkably successfully. The panel additionally commends the College Board for helping to bring together college faculty and high school teachers in collaborative efforts to improve the teaching of calculus across the secondary and postsecondary levels. These efforts have paid off in more coherent syllabi that are better aligned with what is known about the ways in which students learn mathematics. While the IBO’s collaborations with IB teachers are an important aspect of the development of IB programs and assessments, collaborations outside of the IBO network, for example, with professional disciplinary societies or colleges and universities, are not currently part of the approach taken to program development. As the IB program becomes more prominent in the United States, the panel encourages the IBO to expand its collaboration efforts to include more members of the U.S. mathematics community. These types of collaborative efforts can increase public awareness of the program and its relationship to the calculus typically taught in U.S. high schools, colleges, and universities. The remainder of this report is organized into five chapters. Chapter 2 provides an overview of the AP and IB programs in mathematics. Chapters 3 and 4 present, respectively, the panel’s analyses of curriculum and assessment and of teacher preparation and professional development in the two programs. The impact of the programs is examined in Chapter 5. Throughout these four chapters, the panel’s key findings are presented in italic type. Finally, the panel’s recommendations are given in Chapter 6.

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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 2 Overview of the AP and IB Programs in Mathematics Although the AP and IB programs serve similar populations of highly motivated high school students, the College Board and the IBO developed their programs for different purposes. These different purposes affect how the individual courses are developed, structured, and implemented; how student learning is assessed; and how assessment results are used. They also affect how teachers who are assigned to teach AP and IB courses are prepared and supported by the programs and what types of school resources must be allocated for the programs. Detailed information about the history, mission, goals, and growth of the AP and IB programs is available from the College Board and the IBO (see the organizations’ Web sites10). This report includes only information directly relevant to the panel’s analysis, findings, and recommendations. A more complete discussion of the AP and IB programs can be found in the parent committee’s report (Chapters 3 and 4, respectively). THE ADVANCED PLACEMENT PROGRAM Since its inception in the 1950s, the goal of the AP program has been to offer interested, motivated, and well-prepared students the opportunity to tackle college-level material and to earn college credit while they are still in high school. To this end, AP courses are designed to be equivalent to college courses in the corresponding subject area. Standardized, nationally administered, comprehensive achievement examinations that are offered annually in May provide a vehicle for students to demonstrate mastery of the college-level material and to earn college credit or advanced placement in upper-level college courses with qualifying scores.11 The AP mathematics program consists of statistics12 and two levels of calculus—AB and BC.13 Both AB and BC concentrate on calculus of a function of a single variable. Consistent 10   www.collegeboard.com and www.ibo.org. 11   The College Board allows colleges and universities to establish their own criteria for awarding credit or advanced placement. See Chapter 2 of the parent committee’s full report for additional information on this issue. 12   Far more AP students enroll in AP calculus courses and take the related examinations than is the case for the statistics course, although the number of statistics students has been growing steadily since the course was first offered in the fall of 1997. To date, the College Board has identified 22,749 students who have taken both the AP Statistics and the AP calculus examinations; 54 percent took both examinations in the same year, 29 percent took the calculus examination first, and 17 percent took the statistics examination first (Gloria Dion, personal communication). Students who took both examinations earned slightly higher calculus grades than statistics grades.