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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools
Content Panel Report:
Mathematics

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1
Introduction
The National Research Council’s 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.1 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 (for biographical sketches, see Appendix B).
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 Standard Level (SL),2 Mathematics Higher Level (HL), and Further Mathematics SL.3 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
1
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 [4/23/02].
2
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.
3
The panel did not review Mathematical Studies SL because it is a basic rather than an advanced course.

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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, IBO, 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, 1998; Willingham and Morris, 1986) 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. Information about the AP and IB assessments is also limited. The IBO currently conducts no systematic research addressing the validity,6 reliability,7 or comparability8 of its assessments across administrations. 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
4
Downloaded from www.rvcschools.org in August 2000.
5
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.
6
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.
7
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.
8
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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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. 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, currently are not part of the approach taken to program development. As the IB program becomes more prominent in the

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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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2
Overview of the AP and IB Programs in Mathematics
Although the Advanced Placement (AP) and International Baccalaureate (IB) programs serve similar populations of highly motivated high school students, the College Board and the International Baccalaureate Organisation (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 sites9). 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
9
www.collegeboard.com and www.ibo.org.

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college-level material and to earn college credit or advanced placement in upper-level college courses with qualifying scores.10
The AP mathematics program consists of statistics11 and two levels of calculus—AB and BC.12 Both AB and BC concentrate on calculus of a function of a single variable. Consistent with the purpose of the program, AP courses are designed by a development committee comprised of university faculty, mathematicians, and master calculus teachers to be equivalent to many introductory college calculus courses with respect to the range and depth of topics covered, the kinds of textbooks used, the kinds work done by students, and the time and effort required of students (College Entrance Examination Board [CEEB], 1999a). This approach ensures that students have a smooth transition to college, but also means that any shortcomings in college calculus are likely to be transferred to advanced high school students.
AP courses are designed to represent generic introductory-level college courses. The determination of the content to include in AP calculus courses is based on information gathered from responses to an annual survey of college and university departments that offer general calculus courses in their institutions. It should be noted that the response rate to the survey is rather low (30–40 percent), calling into question the consistency of survey results with actual course syllabi in individual schools and in different types of institutions.
Each year the College Board publishes the Advanced Placement Program Course Description: Calculus, AB and BC (often referred to as the Acorn Book). This booklet includes a topical outline for the two calculus courses,13 sample examinations, and information about how previous AP mathematics examinations have been graded (see, for example, CEEB, 1992,
10
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.
11
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.
12
The AB curriculum is generally regarded as equivalent to one semester or two quarters of college calculus, whereas the BC curriculum is considered the equivalent of one year or three quarters of college calculus.
13
The College Board publishes a separate guide for AP Statistics and for each of its other AP courses.

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1996b, 1999b). Although the booklet is published annually, changes from year to year are usually small, and sometimes there are no changes.14
There are no AP calculus curricula in the true sense of the word; there are only topic outlines. The topic outlines are intended to indicate the scope of the courses, but they do not necessarily suggest the order in which the topics are to be taught, nor do they specify any particular pedagogy.15 Teachers are free to determine the order in which topics are introduced, and they are also encouraged to enrich their courses with additional topics as appropriate. Thus as implemented, the curricula for AP courses can vary from school to school in both sequence and emphasis. The resulting curricula are often more a function of the decisions made by individual teachers than decisions made by a centralized curriculum development group.
The topical outline for AP Calculus BC includes all of the topics from AP Calculus AB plus additional topics (see Appendixes C and D, respectively, for the AB and BC topic outlines for May 2000/May 2001). AB and BC students are expected to understand topics common to both courses at the same conceptual level. Both AB and BC calculus are designed to be taught over one school year.16 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 1968, 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. Thus, the program incorporates elements of several national systems without being based on any particular one. IB courses do not reflect the content and structure of college courses but rather what is believed by the program developers to be the full range of content knowledge and skills that should be mastered by well-prepared students prior to
14
The course guide also alerts teachers to upcoming changes so they can prepare to teach the curriculum when it is implemented.
15
A teacher’s guide for AP calculus is available. This guide contains sample syllabi, lesson plans, and activities that have been used successfully by master AP calculus teachers in their own classrooms.
16
The AB curriculum, although given only a semester of credit in college, is almost always taught over a full year in high school. The BC curriculum, which receives a year of college credit, is sometimes spread over more than a year in high school, beginning after one semester of precalculus.

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study at the university level. Thus the courses are developed not to replicate college courses but to reflect an international conception of a well-designed college preparatory program of study among a small group of master teachers and discipline experts who design the courses using both their expertise and information gathered systematically from IB teachers in every IB school. A curriculum review committee with international membership is responsible for articulating, implementing, and maintaining the vision in each subject area. The mathematics committee is also responsible for specifying the presumed knowledge and skills (prerequisites) for each of the four IB mathematics courses.
The IB mathematics curriculum is substantially broader than the AP calculus curricula. IB offerings include four mathematics courses: Mathematical Studies Standard Level (SL), Mathematical Methods SL, Mathematics Higher Level (HL), and Further Mathematics SL. Each course has been designed to meet the needs of a particular group of students, and schools and teachers are encouraged to exercise care in selecting the appropriate course for each student.17 Courses that are designated as SL require a minimum of 150 hours of teacher contact time, while those designated as HL require a minimum of 240 hours of teacher contact time. As noted earlier, 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 (IBO, 1993). For example, Further Mathematics SL is the most advanced mathematics course in the IB program. Both SL and HL courses include a common core of compulsory topics (105 hours for the SL courses and 195 hours for the HL courses), as well as one or more optional topics that are selected by individual teachers for their classes (45 hours for both HL and SL courses). Because the IB program has an international mission, this flexibility in curriculum enables schools to satisfy higher education entrance requirements in many countries.
The IB Mathematics HL course is taught over two years and includes as core material a substantial amount of calculus, roughly equivalent to the AP Calculus AB course, as well as substantial treatment of probability, algebra and trigonometry, complex numbers, mathematical induction, vectors, and matrices. In addition, the HL curriculum has optional units, one of which is taught in addition to the common core. There are units on abstract algebra, graphs and trees, statistics, analysis and approximation, and Euclidean geometry and conic sections. With the analysis and approximation option, the calculus coverage becomes roughly equivalent to that of AP Calculus BC.
17
Mathematical Studies is taken primarily by students who do not expect to pursue mathematics or science in college. Students intending to major in science, mathematics, or engineering usually take Mathematical Methods SL or Mathematics HL, which serve a population comparable to that served by AP Calculus.

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With the statistics option, the statistics coverage becomes roughly equivalent to the purely mathematical parts of AP Statistics (note, however, that AP Statistics devotes substantial attention to the nonmathematical aspects of statistics).
The IB program guides for the mathematics courses present the common aims and objectives for all Group 5 (mathematics) courses (see Box 2-1). The guides also provide a syllabus outline for each course (see Boxes 2-2 through 2-4). In addition to the syllabus outline, IB teachers are provided with a detailed syllabus that defines more explicitly what will be covered on the final examinations. The detailed syllabi include estimated teaching hours for each topic, but do not recommend a sequence for the presentation of topics (see Appendix E for an example of a detailed syllabus). IB mathematics program guides provide general guidance on instruction and offer specific suggestions regarding instructional strategies. The teaching notes also include suggestions for linking content to help students see connections, such as linking the study of the second derivative in the further calculus option to the study of exponents and logarithms in the core content. Teachers use the guides as the basis for determining the structure of their own curriculum. Thus as implemented, the curricula for IB courses, like those for AP courses, can vary from school to school in sequence and emphasis. Little is known about the nature of these variations and their effects on student learning and achievement.
Finding: The AP and IB programs are both designed to meet the educational needs of highly motivated and well-prepared students, but the origins, goals, purposes, missions, organizations, and structures of the two programs are very different. These differences contribute to variations in the educational expectations, opportunities, and experiences of students and teachers participating in the two programs.
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, 1999). The development process incorporates the judgments of both disciplinary and psychometric experts. The AP calculus examinations are timed, with about 50 percent of the total time devoted to multiple-choice questions and the rest to free-response problem-

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+ Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration vectors.
• Optimization, both absolute (global) and relative (local) extrema.
• Modeling rates of change, including related rates problems.
• Use of implicit differentiation to find the derivative of an inverse function.
• Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.
+ Geometric interpretation of differential equations via slope fields and the relationship between slope fields and derivatives of implicitly defined functions.
+ Numerical solution of differential equations using Euler’s method.
+ L’Hôpital’s Rule and its use in determining convergence of improper integrals and series.
Computation of derivatives.
• Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions.
• Basic rules for the derivative of sums, products, and quotients of functions.
• Chain rule and implicit differentiation.
+ Derivatives of parametric, polar, and vector functions.
III. Integrals
Interpretations and properties of definite integrals.
• Computation of Riemann sums using left, right, and midpoint evaluation points.
• Definite integral as a limit of Riemann sums over equal subdivisions.
• Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval:
• Basic properties of definite integrals. (Examples include additivity and linearity.)
*Applications of integrals. Appropriate integrals are used in a variety of applications to model physical, social, or economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the integral of a rate of change to give accumulated change or

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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 (including a region bounded by polar curves), the volume of a solid with known cross sections, the average value of a function, the distance traveled by a particle along a line, and the length of a curve (including a curve given in parametric form).
Fundamental theorem of calculus.
• Use of the Fundamental Theorem to evaluate definite integrals.
• Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined. Techniques of antidifferentiation.
• Antiderivatives following directly from derivatives of basic functions.
+ Antiderivatives by substitution of variables (including change of limits for definite integrals), parts, and simple partial fractions (nonrepeating linear factors only).
+ Improper integrals (as limits of definite integrals).
Applications of antidifferentiation.
• Finding specific antiderivatives using initial conditions, including applications to motion along a line.
• Solving separable differential equations and using them in modeling. In particular, studying the equation y′ = ky and exponential growth.
+ Solving logistic differential equations and using them in modeling.
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.
*IV. Polynomial Approximations and Series
*Concept of series. A series is defined as a sequence of partial sums, and convergence is defined in terms of the limit of the sequence of partial sums. Technology can be used to explore convergence or divergence.
*Series of constants.
+ Motivating examples, including decimal expansion.
+ Geometric series with applications.
+ The harmonic series.
+ Alternating series with error bound.

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+ Terms of series as areas of rectangles and their relationship to improper integrals, including the integral test and its use in testing the convergence of p-series.
+ The ratio test for convergence and divergence.
+ Comparing series to test for convergence or divergence.
*Taylor series.
+ Taylor polynomial approximation with graphical demonstration of convergence. (For example, viewing graphs of various Taylor polynomials of the sine function approximating the sine curve.)
+ Maclaurin series and the general Taylor series centered at x = a.
+ Maclaurin series for the functions ex, sin x, cos x, and 1 / (1 – x).
+ Formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, antidifferentiation, and the formation of new series from known series.
+ Functions defined by power series.
+ Radius and interval of convergence of power series.
+ Lagrange error bound for Taylor polynomials.
SOURCE: College Entrance Examination Board (1999a, 1999c, pp. 9–14).

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Appendix E
Syllabus Details, IB Mathematics HL, Core Calculus Material
SOURCE: International Baccalaureate Organisation (1998a, 1998c).

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8 Core: Calculus Teaching time: 50 hours
The aim of this section is: to introduce the basic concepts and techniques of differential and integral calculus, and some of their applications.

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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 Schools33
TO: Secondary School Mathematics Teachers
FROM: The Mathematical Association of America
The National Council of Teachers of Mathematics
DATE: September, 1986
RE: Calculus in the Secondary School
Dear Colleague:
A single variable calculus course is now well established in the 12th grade at many secondary schools, and the number of students enrolling is increasing substantially each year. In this letter, we would like to discuss two problems that have emerged.
The first problem concerns the relationship between the calculus course offered in high school and the succeeding calculus courses in college. The
33
This 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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Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) recommend that the calculus course offered in the 12th grade should be treated as a college-level course. The expectation should be that a substantial majority of the students taking the course will master the material and will not then repeat the subject upon entrance to college. Too many students now view their 12th grade calculus course as an introduction to calculus with the expectation of repeating the material in college. This causes an undesirable attitude on the part of the student both in secondary school and in college. In secondary school all too often a student may feel “I don’t have to study this subject too seriously, because I have already seen most of the ideas.” Such students typically have considerable difficulty later on as they proceed further into the subject matter.
MAA 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) Calculus Examinations of the College Board, or take a locally-administered college placement examination in calculus. Satisfactory performance on an AP examination carries with it college credit at most universities.
The second problem concerns preparation for the calculus course. MAA and NCTM recommend that students who enroll in a calculus course in secondary school should have demonstrated mastery of algebra, geometry, trigonometry, and coordinate geometry. This means that students should have at least four full years of mathematical preparation beginning with the first course in algebra. The advanced topics in algebra, trigonometry, analytic geometry, complex numbers, and elementary functions studied in depth during the fourth year of preparation are critically important for students’ latter courses in mathematics.
It is important to note that at present many well-prepared students take calculus in the 12th grade, place out of the comparable course in college, and do well in succeeding college courses. Currently, the two most common methods for preparing students for a college-level calculus course in the 12th grade are to begin the first algebra course in the 8th grade or to require students to take second year algebra and geometry concurrently. Students beginning with algebra in the 9th grade, who take only one mathematics course each year in secondary school, should not expect to take calculus in the 12th grade. Instead, they should use the 12th grade to prepare themselves fully for calculus as freshmen in college.
We offer these recommendations in an attempt to strengthen the calculus program in secondary schools. They are not meant to discourage the

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teaching of college-level calculus in the 12th grade to strongly prepared students.
LYNN ARTHUR STEEN
President,
Mathematical Association of America
JOHN A. DOSSEY
President,
National Council of Teachers of Mathematics

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