Appendix A
Overview of Panel Findings and Recommendations

INTRODUCTION

A major goal of this project was to understand advanced study as it is currently implemented in the sciences and mathematics. Accordingly, the National Research Council (NRC) charged the committee with examining advanced study in biology, chemistry, physics, and mathematics (with an emphasis on calculus). The largest advanced study programs for high school students in these disciplines are part of the Advanced Placement (AP) and International Baccalaureate (IB) programs. Because the four subjects are so different, four independent panels of experts in these disciplines were convened to advise the committee.

Panel Composition and Charge

The four panels were assembled using the normal process for appointing all NRC committees: a slate of qualified individuals was identified on the basis of recommendations from a variety of sources and submitted to the NRC for approval. Each panel included an educational researcher with a strong base in the discipline, an accomplished university teacher–scholar in the discipline, and a secondary teacher involved with advanced programs in the discipline. A member of the study committee chaired each panel, and a senior member of the committee’s staff provided assistance and expertise at each of the panel meetings. The chair of each panel served as liaison to the committee. The names of the panel members and chairs are listed at the front of this report.

The charge to the panels (reproduced in Annex A-1 at the end of this appendix) included addressing all the major areas under the full committee’s charge, including evaluation of the AP and IB programs in light of what is known about cognition and learning and the nature of the particular disci-



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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools Appendix A Overview of Panel Findings and Recommendations INTRODUCTION A major goal of this project was to understand advanced study as it is currently implemented in the sciences and mathematics. Accordingly, the National Research Council (NRC) charged the committee with examining advanced study in biology, chemistry, physics, and mathematics (with an emphasis on calculus). The largest advanced study programs for high school students in these disciplines are part of the Advanced Placement (AP) and International Baccalaureate (IB) programs. Because the four subjects are so different, four independent panels of experts in these disciplines were convened to advise the committee. Panel Composition and Charge The four panels were assembled using the normal process for appointing all NRC committees: a slate of qualified individuals was identified on the basis of recommendations from a variety of sources and submitted to the NRC for approval. Each panel included an educational researcher with a strong base in the discipline, an accomplished university teacher–scholar in the discipline, and a secondary teacher involved with advanced programs in the discipline. A member of the study committee chaired each panel, and a senior member of the committee’s staff provided assistance and expertise at each of the panel meetings. The chair of each panel served as liaison to the committee. The names of the panel members and chairs are listed at the front of this report. The charge to the panels (reproduced in Annex A-1 at the end of this appendix) included addressing all the major areas under the full committee’s charge, including evaluation of the AP and IB programs in light of what is known about cognition and learning and the nature of the particular disci-

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools pline, identification of major conceptual issues that should serve as curricular foci, means of balancing breadth and depth, interdisciplinary connections, quality of assessments, teaching methodology, comparison with national standards, and preparation of high school students for further study at the college level. The panels were not asked to consider programs other than AP and IB because of time limitations. Each panel met for two 2-day sessions during the spring and summer of 2000. Prior to each meeting, panel members received general information about the AP and IB programs, as well as materials more specific to their disciplines. These materials included curriculum guidelines; questions from final examinations that had been released by the College Board and the International Baccalaureate Organisation (IBO); and, for the science panels, laboratory manuals used in AP and IB science courses. Panel members also examined other information about research on learning, curriculum, assessment, and teacher education and professional development. Copies of salient national reports, such as National Science Education Standards (NRC, 1996) and Principles and Standards for School Mathematics (National Council of Teachers of Mathematics, 2000), were provided as well. Panel members also applied their personal knowledge and experience with the two programs in formulating findings and recommendations. Review and Interpretation of Panel Findings and Recommendations The chair of each panel assumed responsibility for drafting a report and consulting with panel members to incorporate their suggestions and secure their agreement on the report contents. Each panel report underwent an independent, monitored review by reviewers external to the NRC. The chair of each panel then assumed primary responsibility for preparing the panel’s response to review, along with appropriate changes. The panels’ findings and recommendations represent the consensus of the panel members. Reviewers agreed that the findings are well substantiated. The committee acknowledges, however, that different groups of experts might have arrived at somewhat different recommendations regarding desirable changes. In other words, there may be several solutions to some of the problems that were noted. Since the recommendations of the panels are discipline-specific, the committee did not have the expertise to consider each recommendation in detail. However, the panels’ findings were an important part of the evidence used by the committee in analyzing advanced study programs and in reaching agreement on the recommendations presented in Chapter 11 of this report.

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools As might be expected when four groups of disciplinary experts evaluate educational programs in their subject areas, there are both substantial common elements and significant differences among the panel findings. Problems found to be critical in one discipline are not necessarily important in others. Since each panel undertook its work altogether independently of the others, the common elements are particularly worthy of note. Following a review of some of these common elements, the findings and recommendations of the four panels are presented.1 COMMON ELEMENTS IN THE PANELS’ FINDINGS All of the panels find considerable merit in both the AP and IB programs, which provide challenging opportunities for motivated students that might not otherwise be available. The panels also note major deficiencies in both programs that they believe need to be addressed. The AP calculus program is further along in the process of improvement than are the AP science programs. Research on Learning: All of the panels agree that the AP and IB programs are not yet effectively utilizing what is known about how people learn in developing their courses and their assessments. The science panels also note that the programs are not consistent with national standards such as those presented in National Science Education Standards (NRC, 1996). Although the programs emphasize the importance of higher-order learning and thinking, the amount of content to be covered and assessed, particularly in the sciences, tends to encourage rote memorization rather than conceptual learning. The programs do not yet stress the need for teachers to understand and correct students’ misconceptions. The science panels note that the AP program misses important opportunities to promote interdisciplinary connections between mathematics and the sciences and between different sciences. At present, the IB program does a somewhat better job of recognizing these linkages through its Group 4 Project and its internal assessments in the sciences. Examinations: AP and IB curricula are designed to prepare students for successful performance on end-of-course examinations. The content and structure of the examinations, therefore, have a profound effect on what is taught and how in AP and IB classrooms. The science panels agree that the examinations for AP and IB have some deficiencies. Most serious of these is 1   The full report of each panel, including substantial analysis and supporting material, is available online at www.nap.edu/catalog/10129.html. Those interested in a particular discipline should download the corresponding report.

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools that, in the science panels’ judgment, exam questions do not test conceptual understanding adequately, a situation that adversely affects learning to the extent that classroom instruction is guided by the examinations. The mathematics panel indicates that both the AP and IB mathematics assessments require conceptual understanding, but both examinations could focus on assessing such understanding to a greater degree. Making the AP calculus examinations less predictable, for example, would encourage teachers to teach concepts rather than problem types. Evolution of Course Content and Technology: The biology and chemistry courses in the AP and IB programs do not adequately reflect the evolution of these disciplines over the last two decades. The physics courses also include substantial amounts of older material, but the physics panel does not judge this to be a compelling problem relative to other concerns. Technological advances, especially in the application of computers, are causing major changes in all of the disciplines and generating novel instructional possibilities that are not yet adequately reflected in course content. Teacher Preparation: All four panels note that neither the College Board nor the IBO has provided clear expectations for teacher preparation, that qualified teachers are in short supply, and that better teacher preparation is a prerequisite for substantial program improvement. Student Preparation: All of the panels note the potential for inadequate student preparation for advanced study. In some cases preparation time has become compressed to allow time for the advanced programs. Indeed, as the programs have grown, they have affected the entire high school science and mathematics curriculum. Furthermore, a significant number of students are taking AP science courses as the first course in the discipline, particularly in physics. Additional discussion of these issues is presented in Chapters 2 and 10 of this report. The Secondary–Postsecondary Interface: Finally, all the science panels believe there is insufficient cooperation and communication between high schools and universities with regard to structuring secondary advanced study for students who will enroll in advanced science and mathematics courses when they enter college. The mathematics panel, on the other hand, views cooperation and collaboration between high schools and colleges as a strong point of the AP program. The mathematics panel credits the College Board with facilitating communication among all stakeholders with regard to the teaching of calculus. In all disciplines increased cooperation could lead to improvements in the structure of secondary advanced programs, better teacher preparation and professional development, and college course sequences that are appropriate for students having different experiences in high school.

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools BIOLOGY PANEL Principal Findings Curriculum Finding: The AP course outline is not up to date, and it overemphasizes environmental, population, and organismic (EPO) biology at the expense of molecular, cell, developmental (MCD) and evolutionary biology. Although similarly out of date, the IB curriculum achieves a more appropriate balance between the EPO and MCD areas. The AP curriculum should include more on the process of science, including the responsible conduct of research, and the core IB curriculum should include more evolutionary biology. The core curricula of both programs should be updated to include concepts from current areas of rapid progress, such as genomics, cell signaling, mechanisms of development, and molecular evolution. Finding: A major problem with the AP course is that pressure to cover all of biology in less than a year precludes in-depth study and leads to superficial knowledge. In contrast, the IB program allows time for some in-depth study by subdividing the curriculum into core and options, and by allowing 2 years for the Higher Level (HL) course. The AP course needs to include more options, both in the curriculum and on the tests, to make the breadth covered manageable. One solution would be to have two AP courses—one emphasizing EPO and the other MCD biology—both with significant evolutionary emphasis. Finding: Both AP and IB have stated themes around which the courses are theoretically organized. The eight themes of the AP curriculum mix philosophy and content, with some redundancy in the content themes, but appear to be adequate for their stated purpose. In the IB curriculum, there are only four stated themes, which surprisingly do not include two that appear essential—energy transfer and heredity. Themes in both courses are intended to provide integration of different topics, but the extent to which these themes are used in presenting subject matter generally depends on decisions made by individual teachers. Particularly in AP courses, better integration of topics is needed. Finding: Meaningful learning in biology must involve inquiry-based laboratory experiences that require students not simply to carry out a technique or learn a laboratory skill, but also to pose questions, formulate hypotheses, design experiments to test those hypotheses, collaborate to make experiments work, analyze data, draw conclusions, and present their analyses and conclusions to their peers.

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools Finding: There is little evidence of interdisciplinary emphasis in the AP course outline. In contrast, the entire IB program, including its biology component, rests on the importance of interdisciplinary connections in learning. The IB program is exemplary and far superior in this regard. The AP program should consider changes that would promote interdisciplinary learning. Instruction and Professional Development Finding: AP courses and to a somewhat lesser extent IB courses generally rely on the traditional transmission–reception mode of instruction rather than a constructivist model in which students develop their own conceptual framework through inquiry-based, problem-centered active learning (as recommended in the National Science Education Standards [NRC, 1996]). Both programs need to effect changes in the teaching approaches used. Finding: The way the AP and to a lesser extent the IB courses are taught is inconsistent with current knowledge in several ways: rapid-fire course coverage at the expense of depth of understanding; continued reliance on the traditional passive-learner, transmission–reception model of learning; failure to specifically target common known misconceptions; limited use of history as a route to understanding in the context of people and society; failure to keep pace with new technological and instrumentation opportunities, such as learning through computer modeling of biological systems and hand-held data collection and analysis equipment for field work; over reliance on multiple-choice and fill-in-the-blank test questions; and limited experiential and inquiry-based learning in the laboratory, including the “persuasion of peers” phase crucial to the scientific process. In general, the application of research-based learning theory to the design of instruction and assessment is lacking. Finding: A greater emphasis on inquiry-based learning in AP and IB courses might motivate more students to pursue further training in biology and biology-related careers. Finding: Many teachers at the secondary level are unprepared to teach college-level biology with regard to content knowledge, and many schools that offer AP programs do not have the resources to support adequate laboratory instruction. The College Board should evaluate and certify AP schools and teachers in some manner. Finding: More in-service preparation and support are needed, and more attention should be paid to pedagogy in manuals and workshops, particularly for AP teachers.

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools Assessment Finding: AP and IB final examinations primarily measure rote learning when assessing students’ mastery of content knowledge, concepts, and applications. However, in the IB assessment process, evaluation of a portfolio, laboratory notebooks, and other work provides more perspective. The AP exam should include more free-response questions and evaluation of laboratory work, and the examinations for both programs should test more concept knowledge. With regard to application of knowledge to other courses and situations, the AP exam is limited by a lack of interdisciplinary emphasis, while the IB assessments include such applications. As noted above, the AP course and exam would benefit from more interdisciplinary emphasis. Finding: The perceived need for comprehensiveness and the single high-stakes exam of the AP program in particular encourage teachers to promote rote learning in order to cover all the necessary material. Finding: Both the AP and IB examinations emphasize assessment of what is easily measured: rote learning of facts and concepts, rather than what is most highly valued—hierarchically structured conceptual knowledge and understanding of scientific processes. The Secondary–Postsecondary Interface Finding: University-sponsored outreach programs can be a major resource for high school advanced biology programs and should be encouraged. More communication between high schools and universities—in both directions—would be helpful in fulfilling the needs of both institutions and in developing curricula and assessments. Finding: There are many concerns with the use of AP and IB scores for granting of advanced placement. Some top-ranked colleges do not accept either AP or IB credit or both. For a variety of reasons discussed above, some AP and IP biology courses are not of high enough quality to be appropriate for college credit. The AP biology course as presently constituted is too EPO-oriented to be an appropriate substitute for a first-year college MCD-oriented biology course. Finding: Because of the lack of in-depth study in many AP courses, students who place out of first-year college courses may be at a disadvantage later at institutions where the introductory course is taught effectively. The available data on how well the AP courses prepare students for advanced work in the field may be misleading.

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools Recommendations Overall Recommendation: The College Board should certify schools and teachers that wish to offer AP biology courses and should provide suitable training opportunities for prospective AP biology teachers. The College Board should also develop procedures for ongoing assessment of AP programs and teachers through regular sampling of student work; such sampling should also be used for assessment of student achievement in addition to the final examinations. Overall Recommendation: Certification and assessments of both the AP and IB programs by the College Board and the IBO, respectively, should be designed to ensure that changing emphases in standards for teaching, professional development, assessment, and content, as set forth in the National Science Education Standards (NRC, 1996), are being implemented. Teacher preparation and in-service workshops in both programs should place more emphasis on pedagogy—how to facilitate student-centered, problem-oriented, inquiry-based learning—and on recent results of research on cognition and learning. Overall Recommendation: Colleges and universities should be strongly discouraged from using performance on either the AP or IB examination as the sole basis for automatic placement out of required introductory college courses for biology majors and distribution requirements for nonmajors. Curriculum Recommendation: Students should in general not be allowed to take AP biology as a first science course in high school. A prior biology course should be a prerequisite for AP biology, and a prior chemistry course should be strongly urged as well. In schools where the latter is impractical, chemistry should be a corequisite course. Recommendation: Both the AP and IB curricula should be updated to include topics of major current interest in biology, such as cell signaling, development, genomics, molecular systematics, and their evolutionary implications. Recommendation: The AP curriculum should be better balanced, with more emphasis on molecular and cell biology. The IB core topics should include more evolutionary biology. Recommendation: The College Board should seriously consider offering two different AP biology courses, one emphasizing MCD and the other EPO biology, with two corresponding exams. These courses should go into depth in one of these areas of emphasis and present the basics of the other. Both courses should include a strong emphasis on evolution.

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools Recommendation: More curricular flexibility should be built into the AP program so that students can study fewer areas in greater depth than is possible with the current overemphasis on breadth of coverage. Recommendation: The AP program should place more emphasis on laboratory work by developing a new and larger set of innovative, inquiry-based laboratories that conform to the National Science Education Standards (NRC, 1996) and by including more laboratory-based questions on the exam. Enough laboratories should be available so that teachers have the opportunity to select among them according to their interests and those of their students, and the laboratory-related questions on the AP exam should be general enough so that teachers have real flexibility in deciding which laboratories to offer. In addition, the AP program should include a mandatory 1-week workshop on laboratory pedagogy for beginning teachers of AP biology and should provide more ongoing laboratory training for established teachers. Recommendation: Assessments of schools and teachers should include determination of the amount and quality of the laboratory experience being provided. Scheduling of at least one 2-hour laboratory period per week should be strongly urged as a criterion for certification of an AP biology course. Recommendation: The AP program should promote more interdisciplinary activities that relate AP biology to other academic work, as well as local and regional issues. Assessment Recommendation: The AP program should modify its assessment process to incorporate evaluation of laboratory portfolios and other samples of student work prior to the examination. There also should be more questions on the exam designed to test understanding of major concepts and the process of laboratory research, with less emphasis on rote memorization of facts. Recommendation: To provide feedback, the AP program should make individual students’ exam answers available to their teachers after the examinations have been evaluated. The Secondary–Postsecondary Interface Recommendation: More attention should be paid to the interface between advanced high school and college biology teaching. In particular, more communication and collaboration should be encouraged between college and university departments and high school teachers of biology. Col-

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools leges and universities are potential sources of enrichment and resources for high school courses, and college instructors can benefit from the teaching experience of high school teachers. The need for reform is systemic. Like the AP and IB programs, colleges and universities should revise or improve introductory biology courses as necessary to bring them into line with the recommendations made in this report for high school advanced study courses.

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools CHEMISTRY PANEL Principal Findings Finding: The AP and IB final examinations are formulaic and predictable in their approaches and question types from year to year. Thus, with sufficient practice on how to take such examinations and enough drill on major concepts that the examinations are likely to test, students can score well without actually understanding the major concepts associated with the topics being tested. Finding: The AP and IB chemistry courses do not yet recognize the increasingly interdisciplinary nature of modern chemistry; its incorporation of important related fields, such as materials science and biochemistry; and the opportunities presented by such fields to teach related chemical concepts in a contextual manner. Finding: The AP and IB examinations do not reflect recent developments in chemistry and in the teaching of chemistry at the college/university level. Recommendations Overall Recommendation: Advanced study options in high school chemistry should not necessarily be tied to the potential for earning college/ university credit. Overall Recommendation: Advanced study of chemistry at the high school level should provide students with a coherent, rigorous course that promotes further scientific literacy and prepares students to become part of a highly technological workforce, regardless of whether they continue studying chemistry at the college level. Overall Recommendation: Advanced study in chemistry need not be based on AP or IB. Many top high schools for mathematics and science offer alternatives, which should continue to be explored. Where appropriate, college credit can be sought on the basis of passing the placement examination administered by many college or university departments. Curriculum Recommendation: Any high school course in chemistry that is labeled as advanced study, whether it is structured according to an established curriculum and assessment (such as AP or IB) or otherwise, should enable students to explore the chemistry concepts and laboratory practices introduced in the first-year high school course in greater depth and, where appropriate, to conduct some form of research or independent inquiry. Under

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools PHYSICS PANEL Principal Findings Overall Finding: The most important goals of advanced physics instruction are independent of the particular topics studied. Advanced physics instruction should be aimed at generating excitement and enthusiasm for further study in physics, at achieving deep conceptual understanding of the subject matter covered, and at instilling in students the scientific habits of mind that are important for their further education in science. Learning any particular physics subject matter is of lesser importance. Overall Finding: There are far too few data on the long-term outcomes of physics education to allow important decisions about the physics education of large numbers of students to be made with confidence. Curriculum Finding: The study of Newtonian mechanics provides an ideal framework for developing the scientific habits of mind and deep conceptual understanding that are the primary goals of advanced physics instruction. Moreover, familiarity with Newtonian mechanics is universally expected of students who have completed an advanced high school physics program. Finding: The AP Physics B curriculum is too broad for a 1-year course; it allows insufficient time to develop deep conceptual understanding. The IB Physics Higher Level (HL) course covers largely the same material over 2 years—an ideal pace. Finding: The current AP physics curriculum does not encourage the inclusion of interdisciplinary content in AP physics programs. The IB physics program includes interdisciplinary content as an integral part of the IB program. Instruction Finding: More well-qualified teachers are desperately needed for advanced physics programs. With the continued growth of such programs across the nation, there is a severe shortage of qualified individuals to teach them. Finding: The preparation and skill of the teacher are the principal factors that determine the ultimate success or failure of advanced physics instruction. Thorough understanding of the subject matter is a necessary but not sufficient condition for good physics teaching. Teachers must also be trained in the special pedagogy of physics.

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools Finding: Skilled physics teachers continuously diagnose the understanding of their students and change their objectives and strategies as that diagnosis indicates. It is impossible to assess the understanding of students without requiring them to explain their reasoning. Finding: Thorough understanding of the subject matter is a necessary but not a sufficient condition for good physics teaching. Finding: Traditional “cookbook” methods of laboratory instruction, in which students follow narrowly defined procedures to verify well-known principles, have little effect on students’ conceptual understanding. In contrast, substantial improvements in understanding are possible through rigorous, interactive laboratory experiences. Assessment Finding: Current final assessments place too much emphasis on technical problem-solving skills and insufficient emphasis on deep conceptual understanding. For example, very rarely do the scoring rubrics of either AP or IB physics examinations provide for the deduction of credit for incorrect reasoning; correct final results nearly always receive full credit. Also, both AP and IB physics examinations contain too many multipart questions that lead students through the solution instead of requiring students to demonstrate their understanding by solving the problem on their own. On the whole, IB examinations are somewhat more conceptual in nature than their AP counterparts; recent AP examinations are much better in this regard than those of earlier years. Finding: The scoring of written examinations must emphasize the evaluation of student understanding. Therefore, a rigid scoring rubric in which points are awarded for very specific correct responses to small parts of each question is not appropriate because it reduces the reader’s ability to respond to a student’s thinking (both correct and incorrect) not anticipated by the rubric. Finding: The pace of IB physics final examinations is much more leisurely than that of their AP counterparts. In general, AP physics examinations require students to do too much in too short a time. Recommendations Overall Recommendation: Given the scarcity of data on the long-term outcomes of physics education, an effort should be made as soon as possible to follow the progress of physics students over a period of many years.

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools Student Preparation Recommendation: Before enrolling in an advanced physics course in high school, students should have studied the physics that is suggested as a requirement for high school graduation in the National Science Education Standards (NRC, 1996). This requirement can be satisfied with the first year of a 2-year physics program (the approach adopted by the IB program). Recommendation: Before taking advanced physics, students should be fluent in mathematics through the precalculus level. In particular, by the time they are ready to study advanced physics, students should be skilled in algebraic manipulation and have a firm grasp of basic trigonometry. Emphasis should also be placed on the use of proportions to solve problems, estimation skills, the use of international units, and scientific notation (powers of 10). Curriculum Recommendation: All advanced physics programs should aim to develop deep conceptual understanding of the topics studied. It is essential that whatever topics are chosen be addressed in depth, with an emphasis on conceptual understanding rather than on technical problem-solving skills. Students must learn that physics knowledge is built on general principles and gain the confidence to apply those principles to unfamiliar situations. Recommendation: All students of advanced physics should study a nationally standardized one-semester unit in Newtonian mechanics. This unit should have the coverage of current AP Physics C Mechanics (including rotational dynamics), but should not require formal calculus. It should replace the multiple versions currently offered. Recommendation: In a 1-year advanced physics program, students should study only one major area of physics in addition to Newtonian mechanics. Recommendation: The AP Physics B program is too broad and should be eliminated as a 1-year course. Recommendation: Meaningful real-world (laboratory) experiences should be included in all advanced physics programs. There is ample evidence that traditional “cookbook” laboratories do not meet this standard. Recommendation: Advanced courses should have greater interdisciplinary content and make increasing use of cyberspace and information technology. Modern developments in both science and society as a whole indicate that physicists will be increasingly called upon to address problems that cross the boundaries between traditional disciplines. At the same time, the explosion of information technology provides a vast array of possibilities for improving advanced physics instruction. Teachers and administrators

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools should be aware of these developments and help advanced physics programs expand their involvement in both areas over time. Instruction and Professional Development Recommendation: A concerted effort should be made throughout the physics community to contribute to the preparation and ongoing professional development of highly skilled physics teachers. Peer assessment programs should be implemented for certification and continuing assessment of physics teaching skill. Recommendation: The panel strongly recommends that explanations of reasoning be required from the first day of an advanced course so that providing such explanations quickly becomes automatic for all students in whatever they do throughout the course. Recommendation: Information technology should be used to create networks that will enable high school and college faculty and other professionals to share information useful for advanced physics teaching. Assessment Recommendation: The scoring of written examinations must emphasize the evaluation of student understanding. As noted above, a rigid scoring rubric in which points are awarded for very specific correct responses to small parts of each question is not appropriate; rather, the reader should evaluate the student’s response as a whole. A maximum score should be given only for complete and clear physical reasoning leading to the correct conclusions. The recent trend toward increased emphasis on conceptual understanding should continue. Recommendation: The standards for success on final assessments should be raised. The panel believes that if its recommended curriculum changes are implemented, successful students will know the material in the new, more manageable curricula thoroughly. Therefore, the panel recommends high standards of performance on the new final examinations. Recommendation: The panel recommends that sufficient time be allowed for students to complete an entire examination. Final examinations must measure what students know, not how quickly they can recall and apply that knowledge.

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools MATHEMATICS PANEL Principal Findings Overall 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. Curriculum Finding: The AP calculus curricula are largely sound. The recently revised syllabi with more emphasis on conceptual understanding have significantly improved the program, although further change in this direction is desirable. The panel also finds the focus on reasoning to be less than is needed. Finding: The IB mathematics curricula are largely sound. The portfolio requirement, with its emphasis on applications of mathematics, is likely to introduce a focus on modeling that will benefit IB students. However, the calculus section of the syllabi do not place enough emphasis on conceptual understanding. The panel also has some concern that the breadth of the curricula, although an attractive feature of the program, could lead to superficial learning. Instruction and Professional Development Finding: Neither the College Board nor the IBO explicitly articulates in its published materials what it considers to be excellent teaching in mathematics. 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: 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: U.S. teachers have few opportunities to deepen their understanding of mathematics during the school year, and opportunities during the summer, while useful, tend to be disconnected from everyday teaching.

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools Assessment Finding: AP and IB curricula are designed to prepare students for successful performance on end-of-course examinations. The content and structure of the examinations, therefore, have a profound effect on what is taught and how it is taught in AP and IB classrooms. 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 better understanding of the concept of function. However, the exam is still predictable enough for many students to do respectably well by mastering question types rather than concepts. The exam 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. Finding: The IB exam benefits from being more varied than the AP exams. However, a few exam questions are at too low a level as they ask students to perform algorithms specified in the problem. The exam should include more problems that focus on conceptual understanding, and does 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. 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. 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 appear weaker than the AP examinations in this regard. 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.

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools Impact of Calculus Finding: The adequacy of the preparation students receive before taking calculus has an effect on whether they can understand 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 pre-calculus 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 AP calculus that causes this problem, but the ways in which the program are 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. Finding: Most college and university placement and credit practices that are based on student performance on AP or IB calculus examinations are reasonable. 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 (see also Chapter 10, this volume). Recommendations Preparation of Students Recommendation: All calculus taught in high school should be at a college level. Recommendation: All students who enroll in AP calculus should have had at least 4 years of college preparatory mathematics prior to AP calculus.

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools The structure of IB mathematics courses is different, and this recommendation is not applicable to them. Recommendation: Strategies must be developed to ensure that students who enroll in calculus have an adequate background in algebra and trigonometry for subsequent work in mathematics and science. Instruction and Teacher Professional Development Recommendation: On-going professional development opportunities should be improved, expanded, and made available to all AP and IB teachers. Recommendation: Schools that choose to offer AP and IB programs must find ways to encourage all teachers to take part in professional development, perhaps by providing time during the school day rather than on nights and weekends. Recommendation: The College Board should consider developing procedures to certify AP calculus teachers. Assessment Recommendation: The AP and IB examinations should vary more from year to year. If teachers expect that major ideas will be assessed rather than specific problem types, it is likely that instruction will encourage the development of students’ critical thinking and problem-solving abilities. The AP and IB examinations must strike a balance between judging students’ conceptual understanding by asking unfamiliar probing questions, and alarming teachers and students with a strange and unfamiliar test. Recommendation: Both the AP and IB programs should maintain and increase their focus on conceptual understanding in their assessments. Recommendation: To assess computational and procedural knowledge, students should be asked questions that demonstrate their ability to use these procedures in solving complex problems. Recommendation: All students enrolled in AP and IB courses ordinarily should take the relevant external examinations as part of the course requirements. Recommendation: Both AP and IB examinations should place more emphasis on realistic applications, including those in which students must set up the mathematical model.

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools ANNEX A-1 CHARGE TO THE CONTENT PANELS Charge to the Parent Committee and Content Panels: The charge to the committee is to consider the effectiveness of, and potential improvements to, programs for advanced study of mathematics and science in American high schools. In response to the charge, the committee will consider the two most widely recognized programs for advanced study: the Advanced Placement (AP) and the International Baccalaureate (IB) programs. In addition, the committee will identify and examine other appropriate curricular and instructional alternatives to IB and AP. Emphasis will be placed on the biology, chemistry, physics, and mathematics programs of study. Charge to Content Panels: The content panels are asked to evaluate the AP and IB curricular, instructional, and assessment materials for their specific disciplines. Below is a list of questions that the content panels will use to examine the curriculum, laboratory experiences, and student assessments for their specific subject areas. The content panels will use these questions to issue a report to the committee about the effectiveness of the AP and IB programs for educating able high school students in their respective disciplines. In answering these questions, the content panels should keep in mind the committee’s charge and study questions. The panels should focus on the following specific issues in advising the committee: I. CURRICULAR AND CONCEPTUAL FRAMEWORKS FOR LEARNING Research on cognition suggests that learning and understanding are facilitated when students: (1) have a strong foundation of background knowledge, (2) are taught and understand facts and ideas in the context of a conceptual framework, and (3) learn how to organize information to facilitate retrieval and application in new contexts (see, for example, NRC, 2000b). To what degree do the AP and IB programs incorporate current knowledge about cognition and learning in mathematics and science in their curricula, instructions, and assessments? To what degree is the factual base of information that is provided by the AP and IB curricula and related laboratory experiences adequate for advanced high school study in your discipline? Based on your evaluation of the materials that you received, to what extent do the AP and IB curricula and assessments balance breadth of cov-

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools erage with in-depth study of important topics in the subject area? In your opinion, is this balance an appropriate one for advanced high school learners? Are there key concepts (big ideas) 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)? To what degree are the AP and IB curricula and related laboratory experiences organized around these identified key concepts? To what degree do the AP and IB curricula and related laboratory experiences provide opportunities for students to apply their knowledge to a range of problems and in a variety of contexts? To what extent do the AP and IB curricula and related laboratory experiences encourage students and teachers to make connections among the various disciplines in science and mathematics? II. THE ROLE OF ASSESSMENT Research and experience indicate that assessments of student learning play a key role in determining what and how teachers teach and what and how students learn. Based on your evaluation of the IB and AP final assessments and accompanying scoring guides and rubrics, evaluate to what degree these assessments measure or emphasize: students’ mastery of content knowledge; students’ understanding and application of concepts; and students’ ability to apply what they have learned to other courses and in other situations. To what degree do the AP and IB final assessments assess student mastery of your disciplinary subject at a level that is consistent with expectations for similar courses that are taught at the college level? III. TEACHING Research and experience indicate that learning is facilitated when teachers use a variety of techniques that are purposefully selected to achieve particular learning goals. How effectively do the AP and IB curricula and assessments encourage teachers to use a variety of teaching techniques (e.g., lecture, discussion, laboratory experience and independent investigation)? What preparation is needed to effectively teach advanced mathematics and science courses such as AP and IB?

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools IV. EMPHASES The National Science Education Standards (NRC, 1996) and the NCTM Standards (NCTM, 2000) propose that the emphases of science and mathematics education should change in particular ways (see supplemental materials). To what degree do the AP and IB programs reflect the recommendations in these documents? V. PREPARATION FOR FURTHER STUDY Advanced study at the high school level is often viewed as preparation for continued study at the college level or as a substitute for introductory-level college courses. To what extent do the AP and IB curricula, assessments, and related laboratory experiences in your discipline serve as adequate and appropriate bases for success in college courses beyond the introductory level? To what degree do the AP and IB programs in your discipline reflect changes in knowledge or approaches that are emerging (or have recently occurred) in your discipline? How might coordination between secondary schools and institutions of higher education be enhanced to optimize student learning and continued interest in the discipline?