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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 (2002) Board on Science Education (BOSE)Board on Testing and Assessment (BOTA)

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. "3. Analysis of AP and IB Curricula and Assessments." Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools: Report of the Content Panel for Mathematics. Washington, DC: The National Academies Press, 2002.

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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

computations in a new context. Its problems have a “whiff” of application, but they are often jarringly unreal at a deeper level. For example, question #5 of the free-response section of the 1995 AB examination states:

…water is draining from a conical tank with height 12 feet and diameter 8 feet into a cylindrical tank that has a base area 400π square feet. The depth h, in feet, of the water in the conical tank is changing at the rate of (h – 12) feet per minute.

And question #1 on the 1995 BC examination reads:

A particle moves in the xy-plane so that its position at any time t, 0 ≤ t ≤ π, is

given by and y(t) = 3 sin t.

It is difficult to imagine mechanisms that would make liquid or particles behave as described in these problems. In their attempt to make mathematics seem real, items of this sort may in fact contribute to the sense of many students that mathematics is disconnected from reality.

There is also a paucity of problems requiring substantial background development or technical facility for their solution. For example, there are no modeling problems in which students have to construct a function in an unfamiliar context. Students whose encounter with calculus does not include substantial applications and difficult problems are not likely to regard calculus as the immensely useful problem-solving and explanatory tool it in fact is. The panel notes that the portfolio component of the IB program does require students to perform mathematical investigations, extended closed-problem solving, and mathematical modeling.

The panel is concerned that the need to standardize AP has led to a course with the rough edges smoothed out as much as possible. However, a real appreciation of the subject may require experiencing these rough spots. This is the analog of laboratory work for science courses.

In conclusion:

• The AP examinations are closely aligned with the topics included in the related “Course Description for the AP Calculus Program.”

• Students who do well on the AP examinations can be considered fluent in the basic operations and key ideas of calculus.

• The AP examinations are light in their expectations of technical skill (severity of symbolic calculations) and theory (precision in argument).

• The AP examinations do not place enough emphasis on critical thinking, communication, and reasoning.

• The problems on the AP examinations do not appear to be sufficiently difficult to fully assess important skills and conceptual understanding.

• The AP examinations should include questions that ask students to interpret/explain their results.

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 Front Matter (R1-R2) 1. Introduction (1-4) 2. Overview of the AP and IB Programs in Mathematics (5-15) 3. Analysis of AP and IB Curricula and Assessments (16-31) 4. Teacher Preparation and Professional Development (32-37) 5. Impact of the AP and IB Programs (38-44) 6. Recommendations (45-46) References (47-50) Appendix A: Charge to the Content Panels from the Parent Committee (51-53) Appendix B: Biographical Sketches of Mathematics Content Panel Members (54-55) Appendix C: Topical Outline for AP Calculus AB (56-59) Appendix D: Topical Outline for AP Calculus BC (60-64) Appendix E: Syllabus Details, IB Mathematics HL, Core Calculus Material (65-68) Appendix F: Text of a Letter Endorsed by the Governing Boards of the Mathematical Association of America and the National Council of Teachers of Mathematics Concerning Calculus in the Secondary Schools (69-72)

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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 computations in a new context. Its problems have a “whiff” of application, but they are often jarringly unreal at a deeper level. For example, question #5 of the free-response section of the 1995 AB examination states: …water is draining from a conical tank with height 12 feet and diameter 8 feet into a cylindrical tank that has a base area 400π square feet. The depth h, in feet, of the water in the conical tank is changing at the rate of (h – 12) feet per minute. And question #1 on the 1995 BC examination reads: A particle moves in the xy-plane so that its position at any time t, 0 ≤ t ≤ π, is given by and y(t) = 3 sin t. It is difficult to imagine mechanisms that would make liquid or particles behave as described in these problems. In their attempt to make mathematics seem real, items of this sort may in fact contribute to the sense of many students that mathematics is disconnected from reality. There is also a paucity of problems requiring substantial background development or technical facility for their solution. For example, there are no modeling problems in which students have to construct a function in an unfamiliar context. Students whose encounter with calculus does not include substantial applications and difficult problems are not likely to regard calculus as the immensely useful problem-solving and explanatory tool it in fact is. The panel notes that the portfolio component of the IB program does require students to perform mathematical investigations, extended closed-problem solving, and mathematical modeling. The panel is concerned that the need to standardize AP has led to a course with the rough edges smoothed out as much as possible. However, a real appreciation of the subject may require experiencing these rough spots. This is the analog of laboratory work for science courses. In conclusion: The AP examinations are closely aligned with the topics included in the related “Course Description for the AP Calculus Program.” Students who do well on the AP examinations can be considered fluent in the basic operations and key ideas of calculus. The AP examinations are light in their expectations of technical skill (severity of symbolic calculations) and theory (precision in argument). The AP examinations do not place enough emphasis on critical thinking, communication, and reasoning. The problems on the AP examinations do not appear to be sufficiently difficult to fully assess important skills and conceptual understanding. The AP examinations should include questions that ask students to interpret/explain their results.

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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 The word problems used on the AP examinations are limited in scope. The examinations try for a veneer of reality by including problems that appear to be taken from real-life situations (e.g., water draining and particles moving), but better applications are needed. The types of problems included on the AP examinations are somewhat formulaic and predictable from year to year. This creates a situation that encourages teachers to teach problem types rather than focusing on the development of students’ problem-solving skills and mathematical thinking. Research shows that when teachers know what problem types to expect on an examination, their students tend to focus on lower-level problem-solving behaviors. It is the consensus of the panel that the AP examinations would be greatly improved and would encourage better teaching practices if they included fewer predictable problems and more challenging and interesting problems. Finding: The AP examinations have improved under the current syllabi. The effort to promote conceptual understanding by asking nonstandard questions and requiring verbal explanations is excellent. For example, the fact that there is now a wider variety of applications of integration (and not from a prescribed list) encourages students to think about the meaning of an integral. The inclusion of graphing problems involving a parameter focuses attention on the behavior of a family of functions. The variety of representations of a function—by a graph and a table as well as by a formula—promotes a better understanding of the concept of function. However, the examination is still predictable enough for many students to do respectably well by mastering question types rather than concepts. The examination does not include enough problems that focus on conceptual understanding. More problems are needed that involve multiple steps, test technical skills in the context of applied problems, ask for interpretation and explanation of results, include substantial realistic applications of calculus, and test reasoning or theoretical understanding. The panel acknowledges that including problems such as these on AP examinations, while providing important information about student learning and understanding, could increase testing time and add complexity to the scoring process. A careful analysis of the pros and cons of changing the AP examinations along the lines suggested above should therefore be conducted. One possible solution to this dilemma might be to use multiple measures of student achievement that are administered over a period of time to compute a final AP grade, rather than to maintain exclusive reliance on a single 3-hour examination for determining what students have learned over 5 years of mathematics instruction. IB Program In conducting its analysis of the IB assessments, the panel reviewed the May and November 1999 examination papers for Mathematical Methods SL and Mathematics HL, the Group 5 Mathematics Guide (1993), and the 1998 updated course descriptions for Mathematical Methods SL and Mathematics HL, all published by the IBO (International Baccalaureate Organisation 1999a; 1999b; 1999c; 1999d; 1999e;1999f; 1999g; 1999h; 1999i; 1999j; 1999k). The panel’s analysis of the IB examinations suggests that considerable conceptual understanding

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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 is required for students to do well. However, the current level of emphasis on procedural knowledge does not appear to be appropriate given emerging research on the relative importance of conceptual knowledge. It appears that opportunities are missed throughout the examinations to connect procedural knowledge with conceptual knowledge. The panel is concerned that the procedural aspects of the IB examinations could lead to superficial instruction in the mathematics. The mathematical investigations, extended closed-problem solving, mathematical modeling, and mathematical research options for the internal portfolio assignments are a large step in the right direction to mitigate this problem. Technical Skill The overall technical level of the IB examinations is high compared with typical U.S. high school expectations. Some IB examination questions demand levels of technical accomplishment that the AP Calculus BC examination does not attempt. For example, the May 1999 examination for Mathematics HL, Paper 2, contains the following question 8, part (ii): Using the trapezium rule and Simpson’s rule with 6 sub-intervals, evaluate the integral where g(x) is given at seven points by the following table. x x0 = 0 x1 = 1/6 x2 = 2/6 x3 = 3/6 x4 = 4/6 x5 = 5/6 x6 = 1 g(x) 1 0.97260 0.89483 0.77880 0.64118 0.49935 0.36789 Find the error estimate for Simpson’s rule in terms of g(4)(x). When |g(4)(x)| ≤ 6 , determine the number of subintervals required to use Simpson’s rule to obtain a value for the above integral, which is correct to five decimal places. Items in Further Mathematics SL frequently require a high level of computational skill. Research on learning suggests that procedural fluency also influences students’ ability to utilize their mathematical knowledge. As a result, although the panel calls for an increase in conceptual focus, we do not advocate a decrease in the level of computational skill required. Unlike the AP examinations, all IB examinations allow the use of calculators. The calculators must be of a preapproved type, which include graphing calculators, but not those with a computer algebra system. Although the IB examinations require substantial skill with symbol manipulations, they do not require much graphical and numerical skill. Few problems are primarily graphical or numerical. The numerical problems that appear more often concern numerical algorithms rather than use of numerical data.

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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 The algebraic skills required for success on the IB examinations are impressive. The examinations often involve the use of parameters, which increases the difficulty of the resulting algebra. Most computations in the questions in Part 1 require two steps. For example, on the May 1999 examination for Mathematics HL, Paper 1, questions #6 and #12 are the following problems: 6. Find the value of α for which the following system of equations does not have a unique solution. 12. Given f(x) = x2 + x(2–k) + k2, find the range of values of k for which f(x)>0 for all real values of x. The calculus included on the IB examinations is basic and focuses primarily on techniques. The functions do not appear to be chosen to simplify the computations. For example, in question #14 on the May 1999 examination for Mathematics HL, Paper 1, students were given and asked to find the interval where g″(t)>0 . However, the Subject Report coauthored later by the Chief Examiner and two Deputy Chief Examiners states, “the errors resulting from messy answers [for g’(t)] were sad.” (International Baccalaureate Organisation, 1999). The fact that substitutions for integration and techniques for solving differential equations were given explicitly to the students greatly—and in the panel’s opinion, inappropriately—reduced the level of technical calculus skill required. Problems such as the following appear to be typical: Using the substitution or otherwise, find the integral Conceptual Understanding Few questions on the IB examinations are focused explicitly on conceptual understanding. Many problems could be done procedurally if the teachers had taught this way or if students had done enough practice problems. However, there is such a broad range of topics on the examinations that it would be difficult for students to do well without understanding mathematical concepts.

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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 The IB examinations test conceptual understanding indirectly. Many questions require a significant level of understanding to be solved in a practical manner. For example, consider question #16 from the May 1999 Mathematics HL exam, Paper 1: Given that (1 + x)5 (1 + ax)6 = 1 + bx + 10x2 + ... + a6 x11, find the values of a, b ε Z*. It is unreasonable for a student to multiply out the two expressions on the left to solve this problem; in practice, the student needs to understand which terms of the binomial expansions contribute to the unknown terms on the right. Problems requiring this level of conceptual understanding are not uncommon. Very few IB examination problems require interpretation. Some IB questions also lead the students too much. In such cases, the student often does not need to decide what to do, as this is specified by the question; the “question” revolves around whether the student can perform the appropriate procedure accurately. (The integration problem on substitution given above is of this type.) Theory and Proof Few problems on the IB examinations involve anything resembling theory. Paper 2 of Mathematics or Further Mathematics may include problems on induction, as well as some problems requiring students to “prove” a simple statement (for example, to show that cos x + cos 2x is periodic and even). However, many of these “proofs” involve only one step or a computation. Applications and Modeling The questions on the 1999 IB examinations for Mathematical Methods SL and Mathematics HL make almost no attempt to connect calculus with the real world (although the graph theory and statistics questions do feature quasirealistic settings). However, there is an emphasis on mathematical models in the portfolio section, which was initiated in 2000. There were no problems on the 1999 IB examinations for Mathematics Methods SL and Mathematics HL in which students needed to develop a mathematical model. The statistical problems on the examinations were stated in an applied context, but the students were never required to decide what statistical technique was appropriate for that setting. Some problems had unrealistic modeling settings; others had settings that were never involved in the solution. For example, question #20 on the May 1999 examination for Mathematics HL, Paper 1, reads: A particle moves along a straight line. When it is a distance s from a fixed point, where s > 1, the velocity v is given by Find the acceleration when s = 2.

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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 In conclusion: Including more conceptually focused prompts could strengthen IB test items. This shift in focus would increase the likelihood that instruction would assist students in both overcoming common misconceptions and acquiring a deeper understanding of the mathematical ideas. Some IB questions are too focused on procedure. For example, almost all the integrals to be done by substitution have the substitution given explicitly. This has the effect of making the substitution problems “plug and chug.” Instead, students could be asked to (1) decide that the integral should be done by substitution and (2) be able to select an effective substitution. The IB examination could be improved by including more conceptually focused prompts in items designed to assess students’ techniques for computing solutions. The IB examinations could include prompts that demonstrate the ability to use computational procedures in solving complex problems. The panel acknowledges that including problems such as these on IB examinations, while providing important information about student learning and understanding, could increase testing time. However, some IB testing time might be recaptured and used more effectively if the IBO eliminated the easy prompts that are included to make students feel more comfortable in the testing situation and replaced them with the types of items described above. A careful analysis of the pros and cons of including more complex problems on the IB examinations should be conducted before a decision is made about how to accomplish this task. Finding: The IB examinations benefit from being more varied than the AP examinations. However, a few examination questions are at too low a level as they ask students to perform algorithms specified in the problem. The examinations should include more problems that focus on conceptual understanding, and do not include enough problems that test whether students know which algorithm to apply (e.g., integration by substitution), test technical skills in the context of applied problems, ask for interpretation and explanation of results, and include substantial realistic applications. Examination Practices Beyond their impact on curriculum and pedagogy, AP and IB examinations play a critical role in student learning. Students who do not take the examination at the end of an AP or IB course are less likely to have a college-level experience. Virtually all of the students in IB mathematics courses take the end-of-course examination. Although the panel was unable to obtain precise figures, the Director of the North American IBO office estimated that more than 90 percent of IB students take the examination. In contrast, College Board data suggest that nearly 40 percent of the students who enroll in AP calculus courses do not take the exam. It is unclear why this is so. Some students may decide not to take the examination because they expect not to do well. Others may not take it for financial reasons or because they plan to matriculate at a college where the score will not count. Since schools pay the College Board in

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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 June only for the examinations that are taken, many schools allow students to opt out at the last minute. It also is possible that because schools are under increased scrutiny for quality, and AP examinations may be misused as a measure of quality, weaker students in AP courses may be discouraged from taking the examinations, thereby eliminating potentially low scores from the school’s analysis (see Chapter 10 of the parent committee’s report for further discussion of this issue). Finding: Students who do not take the examination at the conclusion of an AP or IB course miss the opportunity to pull the material together for themselves. They also have a negative effect on the experience of other students by making the course appear to be less serious. COMPARISON OF AP AND IB SYLLABI AND EXAMINATIONS Comparison of the AP and IB syllabi and examinations led the panel to make the following observations: Students who take AP Calculus AB or BC and AP Statistics will likely know more calculus and more statistics, but may know less precalculus (particularly vectors), than a student who completes an IB mathematics program. Students who take IB mathematics will likely know more statistics than those who take AP Calculus AB or BC without AP Statistics. The IB program provides more variety in topics than the AP program and appears to provide more quality assurance in the areas of algebra and trigonometry. As yet, however, there are no data on this point. Both AP and IB examinations have little emphasis on modeling. However, the IB examinations reflect even less of an attempt than the AP examinations to connect to the real world. The AP calculus examinations are still fairly predictable, even after the recent revisions to the syllabi. The IB examinations appear to be less consistent in style and content than the AP examinations, leading the panel to wonder whether IB questions are required to undergo less psychometric screening than AP questions. This situation has advantages and disadvantages. There is less predictability in the IB examination in any given year, and some innovative questions are included, such as question #3 on the November 1999 examination for Mathematics HL, Paper 2. Others, such as the question involving benefit from not having been constructed primarily so they would be easy to grade. However, some IB problems suffer from apparently not having had the level of revision typical of AP questions. (An example is question #8 on the May 1999 Mathematics HL exam, Paper 1, whose correct answer, 4/5, can be obtained by the most obvious wrong method—confusing P(A|S), which is given, with P(S|A), which is the answer.) Finding: The problems on the AP and IB assessments are too predictable. This encourages teachers to focus on helping students recognize and solve particular problem types.

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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 A less predictable examination would encourage instruction focused on the development of students’ critical thinking and problem-solving abilities. Finding: Both AP and IB examinations lack good applications and connections to the real-world uses of mathematics. The IB examinations are weaker than the AP examinations in this regard.

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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 Teaching for the 21st Century, 2000). Current professional development models will be insufficient to ensure adequate numbers of well-prepared and well-qualified AP and IB teachers. This panel is united in asserting that professional development for teachers of mathematics, as for all teachers, must be a planned, collaborative, ongoing, and relevant process. It is not sufficient to offer one-time workshops, regardless of their length, nor is it prudent to assume that good teachers do not need ongoing support from professional communities. Burton (2000) reports that many AP teachers feel isolated and unsupported after leaving AP workshops and returning to their home school districts. AP and IB teachers need opportunities to experiment with new ideas about teaching and learning and to receive feedback about their teaching. Of paramount importance is the need for time to reflect on teaching and learning, both individually and with colleagues. We draw support for these observations from the work of Liping Ma, a mathematics educator, whose 1999 book Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States, sheds light on ways in which professional development for U.S. mathematics teachers can be improved. According to Ma, teachers in China report that their profound understanding of fundamental mathematics was often developed after becoming teachers. The main factors that contribute to Chinese teachers' development of their profound understanding of fundamental mathematics include the following: Learning from colleagues Learning mathematics from their students Learning mathematics by doing problems Teaching round-by-round (teaching grades 1–6, or 1–3, 4–5, which develops an understanding of how mathematical concepts build upon each other through the grades) Studying teaching materials intensively Other important differences between U.S. and Chinese teachers include the following: Chinese teachers spend more time preparing than teaching a lesson. Chinese teachers are organized into “teaching research groups.” Chinese teachers learn a great deal of mathematics from their colleagues. They are assigned mentors, with whom they have regular conversations about mathematics. Teaching materials used by Chinese teachers generally consist of a framework, manuals, and a text. The framework outlines the concepts to be taught, and the manual provides the mathematics background for the corresponding textbook. The manual contains a section-by-section discussion of each topic in the textbook, focusing on the following: What is the concept connected with the topic? What are the important points in teaching the topic?

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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 What are the difficult points in teaching the topic? With Ma’s work in mind, the panel encourages the College Board, the IBO, and individual schools and school districts that offer AP and IB courses to plan, support, and provide professional development activities for AP and IB mathematics teachers that focus on increasing teachers’ subject matter knowledge, knowledge of how students learn mathematics, discipline-specific pedagogical knowledge, mastery of new topics or new approaches to the AP or IB syllabi, opportunities to generate and contribute new knowledge to the profession, and access to collegial communities of AP/IB teachers and to opportunities for collaborative work with college faculty. AP and IB calculus teachers are often the best in their schools, and many compare favorably with teachers at any level. However, AP and IB teachers still need ongoing opportunities and incentives for professional development both to promote their understanding of the underlying mathematical content and to provide information about exemplary strategies for teaching the courses. These support systems might take one or more of the forms discussed below. Workshops Prior to Teaching AB/IB. Both AP and IB teachers are offered optional workshops before beginning to teach an AP or IB course, as well as workshops that cover advanced topics for more experienced teachers.23,24 However, a review of selected workshop materials suggests that some of these workshops may be as much administrative as mathematical in nature. The panel’s vision for an effective workshop is a format in which teachers are provided an opportunity to deepen their understanding of mathematics, as well as develop their pedagogical expertise. Teachers’ Manuals/Background Information. The College Board and the IBO publish and distribute teachers’ manuals (Kennedy, 1997; International Baccalaureate Organisation, 1998b; 1998c; 1998d). However, the panel does not find these materials to be optimally designed to promote the type of instruction emphasized in this report. The panel recommends that manuals be produced in collaboration with high school AP and IB teachers, mathematicians, and mathematics education researchers. The documents should be organized around major mathematical ideas of the AP and IB mathematics courses. We recommend that these manuals include sample questions and answers; text on the mathematical context in which a mathematical idea is situated; and the theoretical underpinnings, the common student misconceptions, and the 23   The IB North America (IBNA) Regional Office is responsible for conducting professional development activities for IB teachers during summer weeklong workshops, 3–5 day sessions during the school year, and regional conferences. IB workshops focus on some issues that are not part of AP workshops, including attention to restructuring of the ninth- and tenth-grade curriculum at IB schools, the nature of the IB student, and the use of international examples and illustrations in the curriculum. Most mathematics workshops include time spent preparing teachers to teach an unfamiliar and expanded mathematics curriculum, including working on vectors and matrices, probability and statistics, and the optional topics (statistics, abstract algebra, and further geometry). 24   Experienced teachers, university faculty, and College Board staff conduct AP workshops. Currently, the College Board exercises little oversight over these workshops other than to list them on its Web site. The College Board intends to take a more active role in the development and implementation of high-quality professional development activities (Commission on the Future of the Advanced Placement Program, 2001).

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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 precalculus and calculus teachers. To address the development of mathematical stamina and persistence in students, staff development should include all teachers, from algebra to calculus. Part of staff development involves providing a stronger foundation in content knowledge for all teachers. All mathematics teachers need to understand calculus so that students can begin working on its underpinnings in algebra. More important, however, staff development involves ongoing professional contact among colleagues at a school. School districts need to create working environments in which teachers have work time to meet and discuss, reflect on, and refine instructional practices. School districts should also create mechanisms for teachers to participate in more structured professional development during contract hours. Currently, most teachers are required to participate in these activities, such as AP or IB teacher workshops, before or after school or during the summer, as suitable substitute teachers are not always available or affordable. The need for more on-site staff development is particularly acute in schools that offer AP. The AP examinations help focus instruction, but because they do this so effectively, material that is not on the tests tends not to get taught in the prerequisite courses. These courses generally contain both students who eventually take AP calculus and those who do not. Thus, accommodations are made in prerequisite courses to ensure that all students are prepared to some extent for AP calculus. The AP calculus examinations have therefore become a 3-hour test that measures 5 years of instruction. To the extent that the test is superficial, the previous 5 years of instruction will also be superficial. Staff development opportunities are needed for the staff to reclaim some ownership of the instructional output at a school. Finding: Neither the College Board nor the IBO explicitly articulates in its published materials what it considers to be excellent teaching in mathematics. Finding: The availability of high-quality professional development activities and the establishment of support networks for AP and IB mathematics teachers are crucial to promoting and maintaining excellence in these programs. Finding: Adequate preparation of teachers for courses leading to calculus or other advanced study options is a critical factor in enabling students to succeed in the advanced courses. Finding: U.S. teachers have few opportunities to deepen their understanding of mathematics during the school year, and opportunities during the summer, while useful, tend to be disconnected from everyday teaching.

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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 5 Impact of the AP and IB Programs Enrollment in AP mathematics courses in the United States far outstrips the enrollment in IB mathematics courses at all levels (see Table 5-1). Consequently, AP calculus has had and continues to exert a far greater influence than the IB courses on the secondary and collegiate mathematics courses offered in the United States. We therefore focus here on the impact of AP. TABLE 5-1 Comparison of Participation in AP and IB Mathematics Courses Based on Numbers of Examinations Administered   AP Calculus AB AP Calculus BC AP Statistics IB Mathematical Methods SL IB Mathematics HL IB Further Mathematics SL Number of Examinations 2000 137,276 34,142 34,118 4,068 1,112 16 AP’S IMPACT ON MATHEMATICS IN GRADES 8–12 For most students, goals such as proficiency in problem solving, proof, and application may be more important than calculus. But realizing these goals take time—time that is not available in a rushed curriculum. As we focus on the AP program and make suggestions for its improvement, we also must ask whether those suggestions will improve mathematics for all. On balance, the effect of AP calculus is probably to improve student achievement in mathematics. However, this is not the case for all students. Because calculus requires 4 years of preparation, the AP Calculus program has repercussions throughout high school mathematics curriculum and even into the middle-school curriculum. Thus, the question of impact may be more important in mathematics than in the sciences. In some schools, the emphasis on preparing a classroom of students for AP calculus governs the curriculum for all other students. This impact has both positive and negative aspects. On the positive side, calculus helps counteract pressures to make high school mathematics easier. For example, such pressures have led to efforts by some programs to get

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