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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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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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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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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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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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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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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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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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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4
Teacher Preparation and Professional Development
The College Board publishes teachers’ guides and sponsors professional development activities for new and experienced teachers; however, it does not specify in its program materials the instructional strategies to be used by teachers. The IB 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” (IBO, 2001:8). 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 (IBO, 1997b:23).
Neither program describes the professional qualifications required for teachers of its courses, and neither provides guidance to schools and districts about the level of instructional resources that should be made available to teachers of advanced mathematics. Such resources include, for example, technology, class time, time for collaborative planning and reflection on professional practice, and opportunities for professional development (see below). The panel notes that this lack of specificity is intentional on the part of the IBO. Because IB is an international program offered in many countries with varying resources, the IBO prefers to let schools determine what resources they can make available in the context of their unique situations. The IBO believes specifying necessary resources would create an insurmountable barrier for schools in poor nations or in poor school districts wishing to offer IB. As part of the process used to authorize schools to offer IB, the IBO ascertains what resources the schools have and how they will be used to offer a quality program.22
Both the AP and IB mathematics programs are expanding rapidly, and there is little reason to expect this growth to taper off in the near future. A natural consequence of this growth is an increased demand for well-qualified and well-prepared teachers to staff AP and IB mathematics classrooms. This increased demand comes at a time when there is already a shortage of qualified teachers who are prepared to teach advanced mathematics—a shortage that will be exacerbated by the large number of retirements expected during the next 10 years and a growing turnover rate among U.S. mathematics teachers. The teacher shortage is particularly notable in rural and inner-city schools (National Commission on Mathematics and Science
22
The process of becoming an IB school can take 2 years or more as the IBO evaluates the ability of both the school and the school district to commit resources to providing the administrative structure, faculty, and facilities needed to support offering the IB diploma program.
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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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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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process by which students acquire understanding. (Note that a new AP Calculus Teachers’ Guide was released in 2000, but the panel did not have the opportunity to review it.)
Mentoring Support: Colleagues and the Internet. An essential ingredient in ongoing teacher development is helping every teacher become part of a comfortable and reliable support group. Teachers, especially novice teachers, need a way to have their questions answered. They must feel comfortable sharing their inexperience with content and pedagogy with colleagues who can offer solutions and support. The best support group is usually departmental colleagues, but teachers of calculus are often the only members of their departments who are assigned to teach this subject. Additionally, new teachers often express trepidation about professing their ignorance or inexperience regarding either mathematical content or pedagogical strategies for fear that doing so may compromise their academic position in the school or their assignment as AP or IB teacher. When school colleagues cannot provide appropriate support, district or regional professional groups are an alternative. The College Board, through its regional offices, and the IBO are uniquely positioned to take a leadership role in helping to develop and support these professional communities of teachers
In recent years, teachers have found an alternative on the Internet. The AP calculus and AP Statistics discussion groups on the Swarthmore College Web site (The Math Forum; http://forum.swarthmore.edu/) have been highly successful in supporting beginning teachers. The online discussions have helped teachers deepen their understanding of the course content and improve their instructional techniques, particularly with regard to technology.25
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. The AP Statistics discussion group is characterized by high-level support offered to high school teachers by leading statisticians. For example, the last two presidents of the American Statistical Association regularly offer background information at a fairly deep level to teachers on the topics in the AP curricula. Their responses to teachers’ questions reveal an understanding and recognition of both the teachers’ level of understanding of the content and the difficulties of teaching statistics. Some panel members note that this level of supportive response is not always true of the AP calculus discussion group. Some teachers who teach both statistics and calculus suggest that while they feel very comfortable asking questions in the statistics group, they hesitate to ask questions in the calculus group. This is the case in part because university mathematicians who participate in the discussions sometimes provide abrupt and condescending responses to questions. It is not uncommon for teachers to feel denigrated by the response, which ends their participation in the discussion.
School-Based Support for Teachers of Advanced High School Mathematics.
Instruction for advanced students can be greatly improved by a coordinated effort among the staff teaching at all levels at a school. At a minimum, staff development is needed for
25
The Swarthmore Web site is The Math Forum, http://forum.swarthmore.edu/. Both the College Board and the IBO also sponsor online discussion groups for mathematics teachers.
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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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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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more students to continue in mathematics by making Algebra II a less challenging course. 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). This practice may send otherwise strong students to college with an inadequate background in algebra and trigonometry, which they will need for the study of science and engineering, as well as further mathematics. In addition, to complete 4 full years of mathematics prior to enrolling in AP calculus, students must either take a mathematics course over the summer or start algebra in grade 8.26 In the United States, few students take algebra before the ninth grade.27 Thus, the desire to have as many students as possible ready to take calculus by their senior year can lead to condensing the mathematics sequence in a fashion that creates long-term problems for some students. For example, students may choose to “double up” and take two mathematics courses in one year—usually geometry and Algebra II. Other students elect to go to summer school to learn geometry or Algebra II. Because they are compressed, summer courses often do not provide a solid background upon which to build higher-level concepts. 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 that is not at the college level.28 Many of these students also lack the necessary algebra and precalculus skills for successful calculus learning. Thus, students in these courses, as well as the less-prepared students in AP calculus courses, would be better served by strengthening their precalculus background and starting calculus in college. A solid background in precalculus is an essential tool for later study in the sciences and quantitative social sciences. The panel sees some merit in developing an AP-level precalculus course for these students (the panel notes that many colleges give credit for precalculus courses). Such a course would be more beneficial to these students than taking a less demanding calculus course or taking AP without adequate preparation, as it would help prepare them for the kind of thinking
26
The panel notes that students who take algebra in grade 8 are sometimes taught by teachers who are not certified to teach mathematics. This practice can lead to having students in calculus courses who lack the necessary algebra skills for successful calculus learning. The 1999 Science and Mathematics Indicators (Council of Chief State School Officers [CCSSO], 1999) reveal that 72 percent of middle-school mathematics teachers were certified in mathematics (up 18 percent from 1994).
27
According to the State Indicators of Science and Mathematics Education (CCSSO, 1999), only 18 percent of all seventh- and eighth-grade students in the United States were enrolled in Algebra I while in middle school. Another 25 percent took algebra in grade 9.
28
In September 1986, the Mathematical Association of America (MAA) 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. The statement was published in Calculus for a New Century (Lynn Steen, editor; MAA Notes #8; 1988). The MAA/NCTM statement includes the recommendation “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 panel supports this recommendation. The full text of the statement is in Appendix E.
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a good calculus course should require. This would be the ideal course in which to hold students accountable for remembering their mathematical knowledge acquired from algebra and applying it to precalculus. It would encourage the development of students’ ability to solve more difficult problems. A strong precalculus course would have the effect of reducing the amount of remediation (currently unacceptably large) that goes on in AP calculus and allowing the exploration of more difficult problems within that course.
Finding: The adequacy of the preparation students receive before taking calculus has an effect on whether they can understand calculus or merely do calculus. Without understanding, students cannot apply what they know and do not remember the calculus they have learned.
Finding: Many teachers and schools are under great pressure to compress algebra and trigonometry so they can prepare as many students as possible for calculus. Sometimes students spend too little time mastering the prerequisite knowledge and skills. The performance of many calculus students is undermined by the fact that they do not learn precalculus thoroughly or learn to solve problems and think mathematically. Thus, the rush to calculus may curtail their future options to pursue mathematics, science, and engineering. It is important to realize that it is not the structure or curriculum of the AP calculus courses that causes this problem, but the way in which the program is used.
Finding: There are not enough checks in the system to ensure that students have the prerequisite algebra, trigonometry, and precalculus skills necessary for success in calculus and courses beyond.
Finding: The courses that precede calculus are often designed to help students make a smooth transition to an AP course. The topics and speed of prerequisite courses are determined by what is needed for AP. Even schools that do not offer AP calculus usually teach from books and curricula that are used in other schools to prepare students for this course. Thus, the AP curriculum influences many more courses and many more students than those who take the AP examinations.
IMPACT OF AP AND IB ON COLLEGE CALCULUS PROGRAMS
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. Although there are no data on the numbers of students in each category, anecdotally the result is college calculus classes in which only a few students have not had some calculus before. Students who have not had any calculus may feel at a disadvantage in this situation, although the data do not always support their concerns. It is interesting to note that many students who have had calculus in high school and repeat the course in college are also at a disadvantage because they do not devote sufficient attention to their college course, perhaps wrongly believing that they know calculus more deeply than they actually do.