Learning and Understanding Improving Advanced Study of Mathematics and Science in U.S. High Schools (2002) / Chapter Skim
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Content Panel Report: Mathematics
Content Panel Report: Mathematics
Pages 491-564
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491 Content Panel Report: , th t.
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the panel focused its work on the Advanced Placement (AP) and International Baccalaureate (IB)
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and several studies that examined the subsequent performance of IB students in college.4 The panel found surprisingly few other data on AP and IB on which to base its evaluation.5 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.
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Nor could the panel find studies that examined the effects on postsecondary mathematics programs of the everincreasing 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)
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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.
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. 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.
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. This booklet includes a topical outline for the two calculus courses,~3 sample examinations, and information about how previous AP mathematics examinations have been graded (see, for example, CEEB, 1992, lathe College Board allows colleges and universities to establish their own criteria for awarding credit or advanced placement.
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Decisions about which AP calculus course an individual student should select are generally based on both the school's offerings and the individual's abilities, achievements, and mastery of the prerequisite material. The International Baccalaureate Diploma Programme The IB Diploma Programme, created in 196S, was originally designed to serve a geographically mobile population, primarily children of diplomats, who relocate frequently but do not want to lose the continuity of their educational experiences.
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IB offerings include four mathematics courses: Mathematical Studies Standard Level (SL) , Mathematical Methods SL, Mathematics Higher Level (HL)
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AP AND IB TEST DEVELOPMENT AP Program AP calculus examinations are designed by a development committee comprised of high school teachers and university faculty in consultation with statisticians and psychometricians who attempt to create examinations that meet accepted standards for technical quality (American Educational Research Association/American Psychological Association/National Council on Measurement in Education, 19991. The development process incorporates the judgments of both disciplinary and psychometric experts.
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502 .~ CONTENT PANEL REPORT BOX 2~] international Baccalaureate Mathematics Common Aims and Objectives AIMS · appreciate the international dimensions of mathematics and the multiplicity of its cultural and historical perspectives · foster enjoyment from engaging in mathematical pursuits and develop an appreciation of the beauty, power, and usefulness of mathematics · develop mathematical knowledge, concepts, and principles · employ and refine the powers of abstraction and generalization · develop patience and persistence in problem solving · have an enhanced awareness of, and utilize the potential of, technological developments in a variety of mathematical contexts · communicate mathematically, both clearly and confidently, in a variety of contexts OBJECTIVES Having followed any one of the programmes in group 5 (mathematics)
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l Number and Algebra Functions and Equations Circular Functions and Trigonometry 4 Vector Geometry 5 Statistics and Probability 6 Calculus PART Ill: Options 1 0 hours 25 hours 1 5 hours 1 5 hours 20 hours 20 hours 35 hours Candidates are required to study all the sub-topics in one of the following options as listed in the Syllabus Details. 8 9 7 Statistical Methods Further Calculus Further Geometry PORTFOLIO 35 hours 35 hours 35 hours 10 hours Five assignments, based on different areas of the syllabus, representing the following three activities: ;~ · Mathematical investigation · Extended closed-problem solving · Mathematical modeling SOURCE: Adapted from IBO (1997)
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9 Statistics 10 Sets, Relations and Groups 1 1 Discrete Mathematics 12 Analysis and Approximation 13 Euclidean Geometry and Conic Sections PORTFOLIO hours hours hours 35 hours 35 hours 10 hours Five assignments, based on different areas of the syllabus, representing the following three activities: · mathematical investigation · extended closed-problem solving · mathematical modeling SOU RCE: Adapted from I SO (1 998a)
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Consequently, this portion of the further mathematics SL programme is regarded as having a total teaching time of 140 hours. l Statistics 35 hours 2 Sets, Relations, and Groups 35 hours 3 Discrete Mathematics 35 hours 4 Analysis and Approximation 35 hours 5 Euclidean Geometry and Conic Sections 35 hours PORTFOLIO 10 hours Three assignments, based on different areas of the syllabus, representing at least two of the following four activities: · mathematical investigation · extended closed-problem solving · mathematical modeling · mathematical research SOU RCE: Adapted from I BO (l 997)
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The internal assessment is a portfolio that represents the teacher's formative assessment of students' practical work judged against established assessment criteria.~9 This component is conducted by teachers within the school environment and is moderated externally by the IBO. The Mathematical Methods SL portfolio consists of five assignments, based on different areas of the syllabus, representing the following three activities: mathematical investigation, extended closed-problem solving, and mathematical modeling.
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Information gained in this manner is used in the preparation of future examinations. Like the College Board, the IBO does not conduct systematic research to determine whether particular test items actually measure the cognitive processes they are intended to measure.
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mathematics programs for advanced high school students, the panel considered the degree to which the curricula and assessments of each emphasize four areas listed below. These areas represent the kind of mathematical learning the panel believes advanced mathematics students should experience in their high school calculus course: · Technical skill, including flexibility with symbolic manipulation, graphical representation, numerical approximation, and function notation · Conceptual understanding, including, for example, the derivative as a rate of change · Theory, including precise definitions, carefully reasoned mathematical arguments, and mathematical rigor · Applications and modeling, including situations in which students must set up the model (for example, choose a formula that represents a real situation)
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IB Program The panel found that a broad factual base of information is provided by the IB curricula for Mathematical Methods Standard Level (SL) and Mathematics Higher Level (HL)
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Finding: 17oeAP curricula are largely sound. 77' e recently revised syllabi with more emphasis on conceptual understanding havesignificantly improved
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Since the types of problems that are on the end-of-course assessments largely determine the structure and content of classroom and homework problems, evaluation of the effectiveness of the programs must include an evaluation of the problems on the examinations. AP Program On the basis of the four areas of emphasis described above, the panel believes the AP examinations should include questions that emphasize the following: ..~ Complex problems that may require significant technical skills · Problems that probe students' conceptual understanding and take into account what research tells us about common misconceptions · Problems that require mathematical reasoning, clear exposition, and the ability to write precise mathematical statements · Problems in which students must construct a mathematical model from a verbal description The panel reviewed 5 years of publicly released AP Calculus AB and AP Calculus BC examinations, related course descriptions, teachers' guides, and scoring rubrics (College Entrance Examination Board ICEEB]
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. This review focused primarily on the examinations that were administered after 1998, when the revised AP calculus courses were fully implemented.
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Conceptual Uncierstanc~ing In recent changes to the syllabi for AP calculus, the emphasis on calculus as a collection of techniques has decreased, and the emphasis on conceptual understanding of fundamental principles has increased. The panel applauds this change.
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Theory anc' Proof Theoretical questions do not appear explicitly on AP examinations; there is little emphasis on proof in the AP syllabi. In the last decade, however, the emphasis on conceptual understanding has increased in the examinations.
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The panel notes that the portfolio component of the IB program does require students to perform mathematical investigations, extended closedproblem 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.
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Me examination does not include enough problems thatfocus on conceptual understanding. Moreproblems 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.
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, 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)
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~ O for all real values of x. The calculus included on the IB examinations is basic and focuses primarily on techniques.
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Conceptual Uncierstancling 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.
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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)
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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 substitutions, 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.
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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.
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MATHEMATICS Finding: Both UP 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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mathematics program guides provide general guidance on instruction and also offer specific suggestions about instructional strategies. Teaching notes for each topic in the syllabus for each mathematics course provide suggestions for teachers while stating "it is not mandatory that these suggestions be followed" (International Baccalaureate Organisation [IBO]
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We draw support for these observations from the work of Liping Ma, a mathematics educator, whose 1999 book Knowing and Teaching ElementaryMathematics: Teachers' Understanding of FundamentalMathematics in China and the United States, sheds light on ways in which professional development for U.S. mathematics teachers can be improved.
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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.
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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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The online discussions have helped teachers deepen their understanding of the course content and improve their instructional techniques, particularly with regard to technology.24 In both discussion groups, more experienced high school teachers offer new teachers suggestions for instruction, help clarify content issues, and suggest references and resources to improve instruction. In addition to secondary teachers helping each other, the discussion groups have regular contributions from university faculty.
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Finding: The availability of high-quality professional development activities and the establishment of support networks forAP 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 criticalfactor in enabling students to succeed in the advanced courses.
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The need to get students ready for the AP courses helps counter such pressures. On the negative side, precalculus courses are condensed in some schools so that three semesters can be set aside to cover AP calculus (especially if the school plans to offer AP Calculus BC)
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High school teachers are aware of the dangers of shortchanging precalculus but are often powerless to resist the pressure from students and parents who believe that calculus on students' transcripts will boost their chances for admission to college. The idea that a large proportion of high school students should reach the level of calculus has led to many schools offering, and a significant number of students taking, non-AP and non-IB calculus courses calculus 25The panel notes that students taking algebra in grade 8 are sometimes taught by teachers who are not certified to teach mathematics.
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and the National Council of Teachers of Mathematics (NCTM) released a joint statement advocating that calculus courses taken in high school be at the college level, in other words, at the level of AP or IB.
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IMPACT OF AP AND IB ON CO' :I~GE CALCULUS PROGRAMS Hi: Although the intent of AP courses is to give students a head start on college-level work, many students take high school calculus and then repeat calculus in college. This group includes students who score well on the AP assessment but retake the course in college in the hope of receiving an A, students who take the AP course but do not take the examination, and students who take non-AP calculus courses.
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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 Entrance Examination Board [CEEB]
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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.29 Neither advanced placement nor acceleration is a goal of the IB program. The International Baccalaureate Organisation does not actively encourage colleges and universities to accept IB examination results for college credit and/or advanced placement in upper-level mathematics courses.
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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. 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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5. The College Board and the International Baccalaureate Organisation should ensure that adequate professional development opportunities are available to all AP and IB teachers.
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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 [4/23/02]
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. College and university guide to the Advanced placement program.
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. International baccalaureate mathematical methods standard level May 1999 exam—M99/520/S(1-2)
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. International baccalaureate mathematical methods standard levelNovember 1999 exa~N99/520/S(1-2)
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. Four years later: A longitudinal study of Advanced placement students in college (College Board Report No.
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and the International Baccalaureate (IB) programs.
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of your discipline around which factual information and ideas should be organized to promote conceptual understanding in advanced study courses (e.g., Newton's laws in physics)
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V PREPARATION FOR FURTHER SIIJDY :~ Advanced study at the high school level is often viewed as preparation for continued study at the college level or as a substitute for introductorylevel college courses.
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She is the coordinator for the MAA Special Interest Group, Research in Undergraduate Mathematics Education, and has been active in various professional organizations, including PME, PME-NA, National Council of Teachers of Mathematics (NCTM) , Mathematical Association of America (MAA)
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He received his Ph.D. in Mathematics Education from North Carolina State University where he completed his doctoral work in international comparisons of calculus education.
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for sample syllabi.) Although the examination is based on the topics listed in the topical outline, teachers may wish to enrich their courses with additional topics.
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· Relationship between the increasing and decreasing behavior of f and the sign off'. · The Mean Value Theorem and its geometric consequences.
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Whatever applications are chosen, the emphasis is on using the integral of a rate of change to give accumulated change or using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region, the volume of a solid with known cross sections, the average value of a function, and the distance traveled by a particle along a line.
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SOURCE: College Entrance Examination Board (1999, pp.
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Although the examination is based on the topics listed in the topical outline, teachers may wish to enrich their courses with additional topics.
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· Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem)
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· Computation of Riemann sums using left, right, and midpoint evaluation points. · Definite integral as a limit of Riemann sums over equal subdivisions.
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Numerical approximations to definite integrals. Use of Riemann and trapezoidal sums to approximate definite integrals of functions represented algebraically, geometrically, and by tables of values.
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+ Lagrange error bound for Taylor polynomials. SOURCE: College Entrance Examination Board (1999a, 1999c, pp.
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557 Ajax E By la us Details' IS Mathematics HL' Core Calculus Material SOURCE: International Baccalaureate Organisation (1998a, 1998c)
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l MATHEMATICS , in, C, ~ is, ~ Ale TO I ~ I I ~~ ' I!
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The first problem concerns the relationship between the calculus course offered in high school and the succeeding calculus courses in college. Ibe 33This letter was extracted from Calculus for a New Century: A Pump, Not a Filter, A National Colloquium, October 28-29, 1987 (MAA Notes, Number 8, edited by Lynn Arthur Steen for the Board on Mathematical Sciences and the Mathematical Sciences Education Board of the National Research Council, Mathematical Association of America, 1988)
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MM and NCTM recommend that all students taking calculus in secondary school who are performing satisfactorily in the course should expect to place out of the comparable college calculus course. Therefore, to verify appropriate placement upon entrance to college, students should either take one of the Advanced Placement (AP)
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DOSSEY President, National Council of Teachers of Mathematics a: 563
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