Notice that question 1 is purely conceptual and requires only a knowledge of the fundamentals of simple circuits. Question 5 probes the students’ ability to deal with the same concepts, now presented in the conventional numerical format. It requires setting up and solving two equations using Kirchhoff’s laws. Most physicists would consider question 1 easy and question 5 harder. As the result in Figure 1.2 indicates, however, students in a conventionally taught class would disagree.
Analysis of the responses reveals the reason for the large peak at 2 for the conceptual question: Over 40% of the students believed that closing the switch doesn’t change the current through the battery but that the current splits into two at the top junction and rejoins at the bottom! In spite of this serious misconception, many still managed to correctly solve the mathematical problem.
Figure 1.3 shows the lack of correlation between scores on the conceptual and conventional problems of Figure 1.1. Although 52% of the scores lie on the broad diagonal band, indicating that these students achieved roughly equal scores on both questions (±3 points), 39% of the students did substantially worse on the conceptual question. (Note that a number of students managed to score zero on the conceptual question and 10 on the conventional one!) Conversely, far fewer students (9%) did worse on the conventional question. This trend was confirmed on many similar pairs of problems during the remainder of the semester: Students tend to perform significantly better when solving standard textbook problems than when solving conceptual problems covering the same subject.
This simple example exposes a number of difficulties in science education. First, it is possible for students to do well on conventional problems by memorizing algorithms without understanding the underlying physics. Second, as a result of this, it is possible for a teacher, even an experienced one, to be completely misled into thinking that students have been taught effectively. Students are subject to the same misconception: They believe they have mastered the material and then are severely frustrated when they discover that their plug-and-chug recipe doesn't work in a different problem.
Degree to Which AP Physics Courses Are Organized Around Key Concepts to Promote Conceptual Understanding
As discussed in Chapter 2, Newtonian mechanics should provide the conceptual foundation for all advanced physics programs. Both AP courses have a substantial mechanics component, but AP Physics C Mechanics, with its coverage of rotational dynamics, is closer to the Newtonian mechanics foundation recommended for all advanced high school physics programs in Chapter 2. On the other hand, we believe that the current AP Physics C Mechanics curriculum contains excessive mathematical complexity that should be eliminated in favor of increased emphasis on conceptual understanding.
Degree to Which the AP Physics Curriculum and Related Laboratory Experiences Provide Opportunities for Students to Apply Their Knowledge to a Range of Problems in a Variety of Contexts
The AP curriculum provides ample material for problem solving. Indeed, problem solving is an indispensable part of any advanced physics course. However, the range and variety of contexts of problems can vary substantially among specific implementations at different high schools. Therefore, the opportunities advanced students have to apply their knowledge to problems in a variety of contexts depend upon the particular AP Physics program in which they are enrolled.
Extent to Which the AP Physics Curriculum and Related Laboratory Experiences Encourage Students and Teachers to Make Connections Among the Various Disciplines in Science and Mathematics
The connection between physics and mathematics is very strong, and both AP Physics courses call upon students to use their mathematical skills to the fullest. Indeed, it is not unusual for students to say after they have taken AP Physics that they appreciate the great value of mathematics for the first time.
On the other hand, the AP Physics program does not encourage students and teachers to connect physics with other areas of science. Rather, such connections are made at the discretion of individual teachers or high schools. These decisions may be shaped by such factors as opportunities for departments to interact or plan courses together, or by the textbooks selected for the course.
Laboratory experiences can indeed be used to make interdisciplinary connections. Yet because the nature of laboratory experiences varies substantially among the implementations of AP Physics at different high schools, it is impossible to answer this question for the AP Physics program as a whole. For a thorough discussion of the importance of laboratory experiences in advanced physics study, see Chapter 2.
Extent to Which Final Assessments in AP Physics Measure or Emphasize Students’ Mastery of Content Knowledge
The coverage of topics on the Physics B examination is excessively broad. The coverage of topics on the Physics C examinations is appropriate, although the nature of the questions (discussed below) may encourage instructional approaches that emphasize mathematical technique rather than conceptual understanding. In that case, students may experience the course as a nonhierarchical litany of facts and formulas (Eylon and Reif , 1984)—effectively too broad and shallow.
On both AP Physics B and C examinations, the standards for success are low. Indeed a 5, the highest possible score, can be obtained for a score of about 60 percent of the total points
available, and a 4 is sometimes given for scores under 40 percent. The College Board justifies this as a way of allowing students to skip examination questions on topics they may not have covered in class. This justification appears plausible for AP Physics B, with its very broad curriculum; however, it is difficult to see how this reasoning can be applied to AP Physics C, with its much narrower curriculum. Allowing students to skip unfamiliar questions on the examinations can be viewed as a means of permitting instructors who are so inclined to pursue fewer topics in greater depth. The panel members’ experiences in preparing students for high-stakes examinations lead us to suspect, however, that few instructors adopt this strategy (this is a question for study), and that the breadth of the examinations is strong motivation for breadth of coverage in instruction.
If the curriculum changes recommended in Chapter 2 were adopted, student mastery of content knowledge could be accurately assessed with improved examinations. At present, the need for the examination to reflect the broad content of the curriculum accurately and consistently, coupled with the low standards discussed above, means that mastery is not being assessed. It is the nature of high-stakes testing that if the test does not assess mastery, teachers will not teach mastery, and most students will therefore not gain it.
Extent to Which Final Assessments in AP Physics Measure or Emphasize Students’ Understanding and Application of Concepts
Many of the questions on previous AP Physics examinations value technique over conceptual understanding. However, the trend on recent AP Physics examinations has definitely been in the direction of increasing the emphasis on conceptual understanding—as it should be. Examples of problems taken from previous AP examinations are presented below. We also note that because the entire final assessment in AP Physics is by means of a single written examination, it is not possible to assess the laboratory skills of AP Physics students (see Chapter 2).
In general, AP Physics examinations require students to do too much in too short a time. Either the time should be lengthened or the amount of material reduced so that students can complete the entire examination. The time issue is important not only because it affects the reliability of the assessment, but also because it influences the instructional practices used in preparing students to take the examination. Preparing students to answer many questions in a short amount of time may not be conducive to instilling in them good scientific habits of mind (see Chapter 2).
The following subsections present four sample problems taken from actual AP Physics examinations (CEEB, 1994a, 1994b, 1999b). The examples taken from the 1993 exams are the kind of technical problems of which we have been critical, while those taken from the 1998 exams are good problems that emphasize conceptual understanding.
A key issue in these examples is the role of mathematics in good free-response questions. We stress that there are two very different aspects of the use of mathematics in physics problems. First is the translation of parts of physical reality into mathematical models that are usually
expressed in the form of equations. Skill in this kind of translation is an important indicator of deep conceptual understanding of physical principles, and its assessment with examination questions is entirely appropriate. Second is the manipulation of the mathematical models to arrive at final results. Skill in this use of mathematics certainly has value, but has little to do with the conceptual understanding of physics. Good mathematical examination questions should therefore emphasize skill in the translation of physical reality into mathematics and downplay the subsequent mathematical manipulations. The panel’s remarks about the examples given in the following subsections should be viewed in light of this principle.
The reader may ask why we do not also present sample multiple-choice questions. It could be argued that, especially since calculators are not permitted on the multiple-choice portion of the AP examinations, many multiple-choice questions are conceptual in nature. However, the panel believes a reliable assessment of the understanding of students can be made only by requiring students to explain their reasoning and then carefully evaluating those explanations, as discussed in Chapter 2. Since the panel regards the assessment of understanding as the primary goal of examinations, the examples given below are taken from the free-response portion of the examinations.
1993 AP Physics B, Problem 2. This problem (see Figure 3-1) calls for the direct use of the formulas for the electric field and potential of point charges, with a small amount of vector addition required in one part. It is more of a mathematics problem than a physics problem. Although important physical principles play some role (superposition of electric fields and the connection between work and potential energy), the problem could be answered correctly by a student who had memorized the relevant formulas and techniques. The problem is thus a poor tool for assessing conceptual understanding.
1998 AP Physics B, Problem 6. This problem (see Figure 3-2) is similar to the questions asked in the Force Concept Inventory (Hestenes, Wells, and Swackhamer, 1992). There is nothing to calculate, yet the problem tests extremely important concepts in projectile motion and Newton’s laws.
1993 AP Physics C Mechanics, Problem 2. This problem (see Figure 3-3) is almost entirely a mathematics problem, involving the solution of an elementary differential equation that many AP Physics students would know from memory. The first two parts contain a rudimentary application of Newton’s second law.
1998 AP Physics C Electricity and Magnetism, Problem 3. This question (see Figure 3-4) is full of important physics. Newton’s laws must be used to express the balance between the component of the weight along the incline and the magnetic force. Faraday’s Law, Lenz’s Law, and Ohm’s Law are all necessary in different parts of the question. Although there is a considerable amount of mathematics involved in part (d), part (e) is a good qualitative question that can be answered from different points of view. We stress that mathematics may be well be necessary to answer some good examination questions, but it is essential that the question focus on important physical concepts.
Extent to Which Final Assessments in AP Physics Measure or Emphasize Students’ Ability to Apply What They Have Learned to Other Courses and in Other Situations
The AP Physics examinations do very little to measure or emphasize students’ ability to apply what they have learned to other fields and situations. The exams are strictly limited to narrowly defined physics content. However, the questions posed sometimes ask students to apply the physical principles they know to situations they may not have previously encountered.
Summary of the Panel’s Evaluation of the AP Physics Program
The AP Physics B program is too broad and should be eliminated as a 1-year course.
The mechanics components of AP Physics B and C should be merged into a single common mechanics component (see Chapter 2).
AP Physics students in a 1-year program should complete the mechanics program and study at most one other major area of physics. They should be examined separately in each area; standards for success should be high.
Examinations should emphasize conceptual understanding rather than mathematical manipulation. Recommendations for improving examinations are given in Chapter 2.
Accomplishing the necessary improvements will require a willingness to alter radically the philosophy of AP Physics examinations. During the transition, the meaning of examination scores will change, so that they may not be consistent with scores on earlier exams. It will therefore be necessary to give up consistency in the statistics of examination scores from year to year as the transition occurs.
THE IB PHYSICS PROGRAM
In this section, we respond directly to the questions under the panel’s charge (see Appendix A) related to curriculum and assessment for the IB Physics program. By IB Physics we mean the Higher Level (HL) IB Physics course, a 2-year course. In contrast to AP Physics, the curriculum for IB Physics is spelled out quite thoroughly by the IBO, down to the number of hours to be spent teaching each rather small division of subject matter.15 For example, 1 hour of teaching time is allocated to the introduction of vectors and scalars, the addition of coplanar vectors by graphical means, and multiplication of vectors by scalars (International Baccalaureate Organization [IBO], 1996, p. 22). In all phases of the course, students are required to spend
approximately 25 percent of their time following an internally assessed scheme of practical/investigative work (IBO, 1996, p. 106).
Schools pay an annual fee to the IBO in order to offer IB courses, and the IBO vigorously enforces its detailed standards for each of its courses offered by a school through periodic inspections and evaluations. IB credit is available only to students who have both completed the IB course and received an acceptable score on the final IB assessment. Again, this practice contrasts sharply with the AP program, in which any student who pays the examination fee can take an AP examination in any subject.
Degree to Which the Factual Base of Information Provided by the IB Curriculum and Related Laboratory Experiences Is Adequate for Advanced High School Study in Physics
There is no question that the factual base of information provided by the IB Physics curriculum and related laboratory experiences is fully adequate for a good advanced program. The factual base for IB Physics is approximately the same as that for AP Physics B, except that the time allotted is a much more reasonable 2 years.
As in the case of AP Physics, the IB program does not mandate the completion of specific laboratory exercises. However, IB Physics students must spend 25 percent of their time doing practical/investigative work “over a prolonged period”16 (IBO, 1996, p. 106). This time need not be devoted entirely to conventional laboratory exercises; computer simulation studies would also be acceptable, for example. However, because teachers of advanced physics courses need to be able to adjust their teaching objectives and techniques in response to their continuing evaluation of the understanding of their students (see Chapter 4), the panel believes the 25 percent requirement is too inflexible.
Extent to Which the IB Curriculum and Assessments Balance Breadth of Coverage with In-Depth Study of Important Topics in Physics
Because the IB Physics program spans a 2-year period, it does not suffer from the same problems as AP Physics B. The material can reasonably be covered in considerable depth in a 2-year course.
IB Physics contains no calculus, and as a result cannot pursue the detailed study of electric and magnetic fields found in AP Physics C. The panel believes that the mathematical study of electricity and magnetism should be an available option in the IB program that could be selected by an instructor with mathematically advanced students.17
Degree to Which IB Physics Courses Are Organized Around Key Concepts to Promote Conceptual Understanding
As noted above, the panel believes Newtonian mechanics should provide the conceptual foundation for all advanced physics programs. IB Physics contains a strong Newtonian mechanics component. In addition, there are already a number of well-defined optional topics that can be selected by individual IB instructors, as recommended in Chapter 2.
The mechanics component of IB Physics currently includes rotational dynamics, as recommended in Chapter 2. However, the new IB Physics syllabus (IBO, 2001), which applies to any student starting the 2-year program in August 2001 or later, does not include this topic.
Degree to Which the IB Physics Curriculum and Related Laboratory Experiences Provide Opportunities for Students to Apply Their Knowledge to a Range of Problems in a Variety of Contexts
The IB curriculum provides ample material for problem solving. Indeed, as noted earlier, problem solving is an indispensable part of any advanced physics course. As with the AP program, however, the range and variety of contexts of problems can vary substantially among specific implementations at different high schools. Therefore, the opportunities advanced students have to apply their knowledge to problems in a variety of contexts depends upon the particular IB Physics program in which they are enrolled.
Extent to Which the IB Physics Curriculum and Related Laboratory Experiences Encourage Students and Teachers to Make Connections Among the Various Disciplines in Science and Mathematics
As noted earlier, the connection between physics and mathematics is very strong, and the IB Physics course calls upon students to use substantial mathematical skills. However, the IB Physics course does not use calculus.
Some of the IB options, such as Biomedical Physics and Historical Physics, connect physics with other disciplines. In addition, all students are required to carry out a Group 4 project in which students are encouraged to combine concepts from different disciplines in a collaborative, investigative experience (IBO, 2001, pp. 27–32). There is great flexibility in the choice of projects and the types of work required; it is therefore difficult to evaluate the educational impact of the Group 4 project requirement.
Extent to Which Final Assessments in IB Physics Measure or Emphasize Students’ Mastery of Content Knowledge
The IB HL Physics final assessment is much more extensive than its AP counterpart. There are three written papers that together account for 76 percent of the student’s final IB score
(given on a 1–7 scale): a 1-hour section consisting of 40 multiple-choice questions; a 2¼-hour free-response section containing one data-based question, several short-answer questions, and one extended-response question; and a 1¼-hour free-response section on the two optional topics studied (IBO, 2001, p. 14). Students have minimal freedom in selecting which questions to answer on the second paper. In contrast, students are required to answer all questions on the AP examinations.
The pace of the IB examinations is much more leisurely than that of the AP examinations, which require a student to complete 70 multiple-choice questions in 90 minutes, and then six to eight free-response problems in a further 90 minutes. Of course, a precise comparison of the time pressure involved in different examinations requires a careful comparison of the questions asked on each exam. Nevertheless, the panel reviewed a considerable number of examinations in each program, and is confident in its assessment that IB examinations create far less time pressure than AP examinations.
The IB examination also allows a 10-minute reading time for written papers. During this time, students can review the paper to decide what questions to answer, but cannot do any written work. The exam time starts after the reading time. The panel applauds the longer time allowed in the IB case; as discussed in Chapter 2, we firmly believe in giving students the time they need to think.
The questions on the IB Physics examinations are generally of a more conceptual nature than those on the AP Physics examinations (as discussed below), another feature the panel applauds. Nevertheless, the written IB Physics examinations suffer from many of the same problems discussed in Chapter 2:
Too many questions have multiple parts that lead students through the solution.
Insufficient attention is paid to reasoning in the scoring rubrics. Too rarely is credit deducted for incorrect statements.
Rigid scoring rubrics need to be replaced by an overall evaluation of the totality of each student’s response.
The standards for success on IB examinations are comparable to those on AP examinations—far too low, in the opinion of the panel.
The remaining 24 percent of each student’s IB score comes from an internal assessment made by the IB teacher of the student’s laboratory work. To make this assessment, the teacher applies a set of stringent criteria that are carefully spelled out by the IBO (2001, pp 15–26). The IBO requires sample papers from schools to verify that these criteria are being applied correctly. Thus the IB final examination provides for an assessment of laboratory skills—something entirely lacking in the AP program. However, the larger question remains of whether the laboratory investigations of IB students are really meaningful educational experiences. If they are not, they are not worth assessing.
Extent to Which Assessments in IB Physics Measure or Emphasize Students’ Understanding and Application of Concepts
As mentioned in the preceding section, IB Physics examinations generally place more emphasis on conceptual understanding than their AP counterparts, as illustrated by the two examples given below. However, this emphasis is still insufficient to enable a reliable assessment of such understanding for the reasons discussed earlier.
Figure 3-5 is question A2 from paper 2 of the November 1999 IB Physics HL examination. There is nothing to calculate, yet the question tests students’ understanding of Newton’s second law in a variety of settings. It is an excellent conceptual question.
Figure 3-6 is question B3 from the same examination paper as Figure 3-5. Part (a) of the problem begins with two “plug-in” uses of the formulas for projectile motion, but then continues with a graph in which students need to demonstrate their understanding of the oscillatory nature of the motion of the ball. The problem then continues with two short essay parts in which students have to explain physical principles. Part (b) is largely an exercise in the use of simple formulas, but part (c) contains another short essay question that tests students’ understanding of angular momentum conservation.
Extent to Which Final Assessments in IB Physics Measure or Emphasize Students’ Ability to Apply What They Have Learned to Other Courses and in Other Situations
The final assessments in IB Physics do little to measure or emphasize students’ ability to apply what they have learned to other fields and situations. Yet they perhaps do somewhat more than the AP examinations in this regard because of the inclusion of optional units that are interdisciplinary in nature.
Summary of the Panel’s Evaluation of the IB Physics Program
The IB Physics program is a good 2-year course. With respect to the pace at which it addresses topics, covering largely the same material as AP Physics B, it may be ideal.
The IB curriculum’s fixed minimum allocations of time to activities such as “practical investigative work” and the Group 4 project are somewhat troubling; it is far from clear that all the mandated activities are educationally worthwhile. Also, such fixed time allocations are in conflict with the teacher’s need to adjust instructional strategy in response to a continually changing diagnosis of student understanding (see Chapter 4).
The IB Physics course contains no calculus and therefore cannot delve as deeply as AP Physics C into electromagnetic theory.
The IB Physics examinations are more conceptually based than the AP Physics examinations and give students sufficient time to think, but suffer from many of the same faults of construction as the AP exams.
Unlike the AP program, the IB program can assess laboratory skills by means of the internal assessment process.
Teaching and Learning in Advanced Physics Programs
In the preceding chapters, advanced high school physics programs are discussed as if they can be separated from the particular teachers and students involved. Adopting this point of view greatly simplifies the discussion and makes it possible to draw some important conclusions. Nevertheless, the panel is convinced that the success of a given advanced high school physics program depends much more on the teachers and students than on the curriculum or other general program characteristics. We stress that in truth, there is no such thing as the AP Physics program or the IB Physics program. Rather, as noted earlier, the implementation of these programs varies widely from school to school and teacher to teacher, and these variations often have a much greater impact on the education of the students than the choice of which program to implement. In this chapter, we examine the characteristics of teachers and students that give rise to these critical variations in the implementations of AP and IB Physics. Based on this examination, we offer several recommendations for improving the overall quality of advanced high school physics instruction.
THE TEACHERS OF AP AND IB PHYSICS
In the opinion of the panel, the most important factors in determining the quality of advanced physics instruction in high school are the talent and preparation of the teacher.18 To achieve widespread high-quality advanced physics instruction, it is necessary to provide for both recruitment of highly qualified college students into the teaching profession and strong, substantive pre-service and in-service professional development of physics teachers.19
Competitiveness of Salaries of Physics Teachers
While both beginning and average teaching salaries have improved relative to inflation since the 1980s (Neuschatz and McFarling, 1999), the present competition for technical expertise makes salary an important factor in attracting new teachers to the profession. It is difficult to see how enough well-qualified college students can be attracted to the teaching of advanced physics courses when a similar amount of training would allow them to earn much higher salaries in other positions.
Low salaries not only make it difficult to attract people to the teaching profession, but also lead to a general lack of respect for teachers. “If you could do something else, you wouldn’t be here” is a far too common conclusion reached by students, parents, and others about the teachers they depend on for education.
Role of AP and IB Organizations in the Training and Credentialing of Teachers
At present, neither the AP nor the IB program requires any special credentials or teacher preparation for teaching its courses other than what is required by the school and the state in which the school is located. The IB program does require teachers to be trained in IB procedures when a school first joins the IB program; this initial training also includes a content workshop for teachers. In addition, the IB program provides numerous optional content workshops for IB Physics teachers at locations around the world.
A number of institutions offer 1- or 2-day AP workshops or summer institutes, but these are not required of teachers offering AP courses. The panel suggests that the College Board and the IBO consider whether requiring such workshops or introducing some form of mandatory or optional credentialing would improve the quality of preparation of teachers of their programs.
A recent American Institute of Physics survey (Neuschatz and McFarling, 1999) notes that about one-third of high school physics teachers have a degree in physics or physics education, and an additional one-sixth have a minor in physics. Almost all the rest have degrees either in another science or in mathematics. Many report being inadequately trained to use computers in the laboratory and the classroom, and only half report taking part in any physics workshop, meeting, or course during the year preceding the survey. The panel believes such training deficiencies are a major cause of unsuccessful outcomes in advanced physics teaching.
help create a scientifically literate public, capable of making informed decisions on public policy involving scientific matters. A strong K-12 physics education is also the first step in producing the next generation of researchers, innovators, and technical workers.
Involvement of College and University Physics and Engineering Faculty in the Training of Teachers of Advanced High School Physics
The National Science Education Standards (NRC, 1996) contain detailed standards for the professional development of science teachers. The panel urges the cooperation of the entire physics community in the recruitment and training of physics teachers who meet those standards. In particular, we join the American Institute of Physics, the American Physical Society, the American Association of Physics Teachers, and other physics organizations in calling on college physics and engineering faculty to take an active role in the training of teachers for advanced high school physics programs.20
The panel concurs with the opinion of Robert Watson, Director of the Division of Undergraduate Education at the National Science Foundation, that if science departments in colleges and universities were more hospitable to students who wanted to become teachers, not only would those students be better prepared to go into teaching, but a much stronger cadre of students would be attracted to teaching (NRC, 1997a, p. 4). Physics departments need to treat high school physics teaching as an honorable specialization for an undergraduate physics major, and provide both undergraduate and graduate courses of study aimed at the preparation and continued professional development of skilled physics teachers.
Physics departments should pose the prospect of high school teaching as an option in advising students, perhaps in conjunction with a local physics teacher who could provide perspective on such a career choice. Physics majors specializing in education should take the same foundational courses as conventional physics majors, replacing some elective or specialized coursework with courses of similar rigor that probe in depth the problems of learning and teaching physics. Such courses should include the use of traditional techniques, as well as modern teaching techniques based on research on physics education. Recognizing that a substantial fraction of all undergraduate physics majors will end up instructing others in some fashion at some time in their careers, a course in teaching and learning would be a useful addition to the physics major in general, although the details of such a course would vary. The panel notes that some programs of this type are already in operation (see for example, Roelofs, 1997).
Preparation Needed to Teach Advanced Physics Courses Effectively
There is no doubt that to teach advanced physics effectively, a strong background in physics content is absolutely essential. Teachers who do not understand the subject themselves cannot possibly develop deep conceptual understanding of physical principles in their students.
However, a thorough understanding of the subject matter is not sufficient. A teacher must also be skilled in the specialized pedagogy of physics. Physics teachers need to be trained to understand their students’ view of the physical world (which is often radically different from that of physicists) and adjust their teaching techniques accordingly (as discussed below).
See the December 1999 Statement on the Education of Future Teachers at www.AIP.org.
EFFECTIVE TEACHING IN ADVANCED PHYSICS PROGRAMS
Another reason for the varying degrees of success among teachers is that different teachers teach in different ways. A great deal of research has been done to identify the most and least effective teaching practices for educating students of science. In this section, we summarize the main results of that research from the point of view of advanced physics instruction.
Findings of Research on How People Learn
A number of summaries of the implications of recent research on learning for the teaching of science have been prepared. The panel notes and endorses the NRC (2000) report How People Learn: Brain, Mind, Experience, and School: Expanded Edition. Adapting and elaborating these findings specifically for the study of physics leads to the following implications for effective instruction.
Teachers must draw out and work with students’ current understandings, including those students bring with them to the course and those they develop as the course progresses.
There is a robust consensus in education research, including a substantial body of work specific to introductory physics (Clement, 1982; Champagne, Gunstone, and Klopfer, 1985; Hake, 1998; Hestenes, Wells, and Swackhamer, 1992; McDermott and Redish, 1999), that to be effective, instruction must elicit, engage, and respond substantively to student understandings. There are now a number of examples of curricula and materials designed to support productive interaction with student understandings (as discussed later in this chapter), and there is evidence that these approaches can achieve progress in understanding not possible for most students with traditional methods.
Teachers must address students’ metacognitive skills, habits, and epistemologies.
Students need to understand not only the concepts of physics, but also the nature of knowledge and learning (Halloun, 1998; Hammer, 1995; Hewson, 1985; McDermott, 1991; Redish, Steinberg, and Saul, 1998; Reif and Larkin, 1991; White and Frederiksen, 1998). Many students arrive at physics courses, including advanced courses, expecting to learn by memorizing formulas disconnected from each other, as well as from the students’ experiences of the physical world. Effective instruction challenges these expectations, helping students come to see physics learning as a matter of applying and refining their current understanding. Students must learn to identify and examine assumptions hidden in their reasoning; to monitor the quality and consistency of their understanding; to formulate, implement, critique, and refine models of physical phenomena; and to make use of a spectrum of appropriate representational tools. By the end of an advanced course, students should have developed a rich sense of the coherent, principled structure of physics, and be both able and inclined to apply those principles in unfamiliar situations. In short, effective instruction should work toward the objectives identified in Chapter 2.
Many teachers start off well by explaining to their students that success in studying physics depends on learning and applying general principles, rather than on memorizing many unrelated facts and formulas. Talk, however, is cheap. Too many teachers follow up these commendable words by evaluating their students on the basis of their skill in solving problems that are handled most efficiently by just such memorization. Teachers must design the assessments in their courses to make the statement that “success depends on learning and applying general principles” a reality.
Effective teachers are sophisticated diagnosticians of student knowledge, reasoning, and participation.
How teachers respond to student thinking depends critically on what they perceive in that thinking, on what they interpret to be the strengths and weaknesses of the student’s understanding and approach.21 Effective teachers continually gather information to support this ongoing assessment from several different sources: written work on assignments, tests and quizzes, classroom discussions, and contact with students outside the classroom. They ask students to explain their reasoning throughout their work. Upon gaining new insights into student understanding, effective teachers adapt their instructional strategies and objectives. For this reason, teachers must be trained both to make accurate diagnoses of student thinking and to devise effective responses to that thinking (as discussed earlier).
Teachers must teach a smaller number of topics in greater depth, providing many examples in which the same concept is at work.
This is a common refrain in findings from education research, often expressed in the slogan “less is more.” In part, this precept is an implication of the previous two: drawing out and working with student understandings and addressing metacognitive skills and habits all take time, and this necessitates a reduction in the breadth of coverage. Education research also yields the robust finding that coming to understand a concept requires multiple encounters in multiple contexts. This finding is reflected across innovations in physics curricula (see the discussion later in this chapter).
A related consideration is that effective teaching requires limiting class sizes to a reasonable number. Drawing out, evaluating, and working to improve student understandings is a highly individualized process that is nearly impossible to perform adequately in oversized classes.
These admonitions from education research are not controversial, but their generality may allow multiple, superficial, and possibly conflicting interpretations. If these points are to be meaningful, it is essential that they be discussed with respect to specific instances of learning and
For an extended discussion and example of diagnosis in a high school physics class, see Hammer (1997). For an account of facets-based diagnosis, see Minstrell (2000). Diagnoser, a software tool designed to support facets-based diagnosis of student thinking, is available at http://depts.washington.edu/huntlab/diagnoser/.
instruction (examples include Minstrell, 1989; Hammer, 1997; Roth and Lucas, 1997; and van Zee and Minstrell, 1997).
Special Considerations in Advanced Physics Instruction
The panel acknowledges that there are students who are successful at learning under almost any circumstances, that these students often appear in advanced courses, and that for them the implications of the previous section may not be as critical. However, such students do not appear to constitute the majority of those enrolled in advanced courses. Moreover, even the best students would benefit from the recommendations of the research discussed above.
With these considerations in mind, the panel’s first concern regarding advanced courses is that the breadth of the curriculum often leaves very little time to do anything but introduce new subject matter. Therefore, a serious attempt to improve instructional practices will require giving up coverage of some material. The panel’s recommendations for accomplishing this are presented in Chapter 2.
The panel’s second concern is that the emphasis in advanced courses on solving typical textbook problems as opposed to conceptual discussion and debate tends to encourage rather than challenge student perceptions that physics knowledge comprises disconnected factual units rather than a principled system of ideas. Although skill in problem solving is an important objective of any advanced physics program, conceptual understanding must be the objective of highest priority.
Extent to Which the AP and IB Curricula and Assessments Encourage Teachers to Use a Variety of Teaching Techniques
As described in Chapter 3, the College Board leaves the way in which AP Physics is taught entirely to the particular implementation of AP Physics in each school. There are no requirements that must be met before students can take the AP examinations. Although the College Board does offer advice to teachers in various publications, it is entirely up to the particular school whether to accept or reject that advice. The only means the College Board has of encouraging teachers to use a variety of techniques is through the construction of the examinations. Recent examinations have tended to emphasize conceptual understanding and real-world situations to a greater degree than earlier examinations. To the extent that a variety of teaching techniques is necessary to prepare students to answer these kinds of questions, the use of such varying techniques is encouraged.
In contrast, the IBO spells out in great detail the time to be spent doing different kinds of activities in IB Physics courses. For example, 25 percent of instructional time is to be spent by students pursuing a program of “practical work” (see Chapter 3). Moreover, a great deal of written material is available to IB teachers that contains detailed suggestions for carrying out these activities. Thus the IBO most definitely encourages the use of a variety of teaching techniques, and in fact goes a long way toward mandating the use of such a variety.
Whether the use of that variety is effective in the IB program, however, is entirely another matter. As discussed earlier, teachers need to be free to adjust their teaching techniques and objectives as they continually diagnose the state of understanding of their students. The IB program’s prescriptions of time to be spent in various activities may deprive teachers of the flexibility they need. The panel is skeptical that such inflexible prescriptions are appropriate for advanced physics instruction.
The panel has been informed through personal testimony and anecdotal evidence that some experienced IB Physics teachers do not feel constrained to follow these time prescriptions. The panel considers the full use of whatever flexibility is available to be entirely appropriate.
Differences Between High School Physics Instruction in the United States and Other Countries
In light of the poor performance of U.S. students on the physics portion of the Third International Mathematics and Science Study (TIMSS) as compared with students from other developed countries (see U.S. Department of Education, 1998), certain characteristics of the U.S. physics education program need serious reconsideration and further study. To be clear, the panel is not endorsing TIMSS as the way to measure success in physics education; rather, we believe the lower scores of American students are a good reason to explore what can be learned from the methods of physics instruction used in other TIMSS countries.
Perhaps the most obvious difference between physics instruction in the United States and other countries is the number of secondary school years devoted to physics study. In most TIMSS countries, students develop their physics knowledge over a period of many years, rather than the traditional 1 year commonly offered in the United States. The system used by other countries appears to be more consistent with the findings of education research that coming to understand a concept requires multiple encounters in multiple contexts.
One possibility for American programs to consider is moving the single year to a different position in the traditional “layer-cake sequence.” Doing so might change the mathematical level at which the physics is taught or might involve students at a different stage of cognitive development. It might also facilitate the use of physics in interdisciplinary high school science classes such as those encompassing biology and physics.
Another alternative is to offer physics along with other scientific disciplines in an integrated or coordinated course for the first 2 years of high school, and then offer further physics study in the junior or senior year at an advanced level. This approach is similar to that of science programs in most comparison countries. Additional years of integrated science courses in the third and fourth high school years is another possibility. Given the scarcity of comparison data on these sequences and the fact that many schools in the United States are making the transition to integrated courses as a result of exciting new curricular offerings, this is an excellent time to examine these issues.
A second significant area of difference between physics instruction in the United States and other TIMSS countries lies in the recruiting, training, professional development, and status of physics teachers. The salary situation for American secondary school physics teachers is particularly troubling (as noted earlier). First, the difference in salary between positions attainable by physics majors (e.g., computer programmer, laboratory technician) and the position of beginning teacher is much greater in the United States than in most comparison countries. Second, the comparison with professions unrelated to physics is no better. In Japan, for example, secondary school teachers are in the top 20 percent of the professional salary scale; in the United States, they are in the bottom 20 percent. Salary issues related to recruitment and retention need to be addressed for the U.S. physics teaching profession before we find ourselves with far too few qualified teachers or mentors.
A third area of difference is reflected in the fact that some of the TIMSS content involves topics not commonly covered in U.S. physics courses. One example, thermodynamics, not only is poorly understood by many physics teachers and therefore avoided, but also is often left to chemistry teachers as a part of their curriculum. Although the United States is generally perceived to teach more science topics in less depth than is the case in other countries, physics topics beyond mechanics and electricity and magnetism are often mentioned in a highly perfunctory fashion. U.S. students may well have no experience with these topics, especially on exams. Other examples are the lack of emphasis on statistics and statistical analysis and the lower factual content in U.S. textbooks compared with many European high school texts. More data are needed to determine whether these and other mismatches put U.S. students at a disadvantage.
U.S. students also appear to spend more class time doing laboratory work and to do more independent work than their counterparts in other countries. Although many believe these uses of student time encourage the development of creative, bold, and imaginative physicists, the time spent in the laboratory may diminish that devoted to other physics class activities, such as problem solving, which may in turn affect test scores.22
Finally, unlike the United States, many comparison countries have extensive evaluation and assessment programs. Since most foreign countries have national educational assessments, their evaluation processes may be more efficient and meaningful in such disciplines as physics. Some programs have been in place for at least 100 years, involve highly trained teachers as external graders, and publish school scores publicly. The availability of long-term student assessment data may be an important factor leading to the educational success of those programs.
THE STUDENTS OF AP AND IB PHYSICS
This section examines how the nature of the student body affects educational outcomes in advanced physics instruction. The students who take AP and IB Physics courses come from a wide variety of academic, cultural, and socioeconomic backgrounds. This diversity can only increase as both programs continue to expand in size. An effective implementation of AP or IB
Physics must take into account the background of the students enrolled in that particular implementation.
The Problem of Widely Varying Levels of Academic Preparation and Ability
Variations in abilities and preparation levels make teaching an advanced physics program more difficult for both teacher and students. When the ability or preparation of some students is far below the expected prerequisite level for advanced study, the teacher is faced with the unenviable choice of either decreasing the extent or depth of content coverage to accommodate those poorly prepared students, thereby failing to meet the needs of the students who are adequately prepared; or teaching the course at the level appropriate for the adequately prepared students, thereby failing to meet the needs of those who are poorly prepared.
The panel wishes to emphasize that preparation must include more than background knowledge and related skills. It must also include the development of mature work habits. Students who do not complete their homework or are generally unwilling or unable to put forth sustained high levels of effort are unlikely to succeed in advanced physics study in high school and may have effects on other students’ learning that cannot be ignored.
It is not possible to avoid inappropriate placement situations entirely—nor would doing so be desirable. Students who want to try a difficult course because they are genuinely interested should be encouraged to do so, provided they are made fully aware of the commitment of time and effort they are about to make. For some of these students, a well-taught and challenging advanced course in physics could spark learning and interest that had previously gone unrecognized. On the other hand, seriously underqualified students should be discouraged from taking advanced physics. Doing poorly in a course for which a student is not prepared often discourages the student from further study in the field and certainly will not enhance his or her application to college.
These are complex issues with no easy solutions, and we recognize that not everyone will share our views. The panel recommends discretion in the assignment of students to advanced courses,23 so that the teacher is able to maintain the integrity of the course as an advanced treatment of the concepts being covered. For example, in solving projectile motion problems, a comprehensive treatment is commonly not accessible to some fraction of the class simply because they do not know how to solve quadratic equations. The panel believes appropriate members of each school’s staff should be involved in decisions about admitting individual students to the school’s advanced physics program. Of course, the school staff should consult with students and their parents and consider their views carefully in the deliberations, as appropriate. School personnel, students, and parents should be clear about what constitutes adequate preparation for an advanced physics program. Such preparation might include background knowledge and related skills, such as mature work habits.
At first glance, it may appear most fair to deny admission to underprepared students. However, many students lack the necessary preparation because of limited educational opportunities beyond their control. Moreover, as noted above, some of these students may experience an academic awakening when provided with such opportunities. The panel believes it is the job of the school staff to identify those underprepared students who are likely candidates for such experiences and admit them to the advanced physics program regardless of prerequisites. Doing so may require extensive interviews with the student, the student’s parents and teachers, and others who know the student well. Making these judgments is not easy, but we believe it is within the capability of a skilled educational staff.
Ultimately, the way to expand access to advanced physics is to reform the education system so that all students will be able to experience a rigorous academic program that will allow them to study advanced physics in their late high school years if they choose to do so.
Role Played by the Culture of the American High School
Students do not take advanced physics in a vacuum. They are constantly interacting with the surrounding high school environment, which places substantial demands on them physically, intellectually, socially, and morally. Since the study of advanced physics is itself a demanding pursuit, there is a constant struggle between the advanced physics program and other activities for the attention of students. To be successful, an advanced physics program must adapt itself to the particular high school environment and find a way to use whatever time students can devote to the program to best educational advantage.
Competing activities that place demands on the time of advanced physics students include part-time employment; athletics; community service; music and drama; college visits; television and movies; computer activities, including Web surfing; family vacations; and social events. Much of the pressure to engage in these activities comes from colleges and universities. In a desire to enhance their college applications, students often undertake an enormous variety of activities that leave them too little time for their studies. Pulled in so many directions at the same time, these students have a low attention span, both in class and in doing their homework.
It must also be recognized that English is not always the native language of advanced physics students. Since the concepts of advanced physics are new and difficult for many students, a poor command of English can cause serious problems for those who are otherwise quite well prepared for the course. Even for native English speakers, the specialized use of common words such as momentum or work often leads to confusion. Those effects are exacerbated among students for whom English is a second language. In addition, cultural influences can affect how students think about science: reasoning by analogy or by strict linear logic; memorization of specific correct responses or generalization; problem solving by induction or by deduction; or needing to learn through hands-on apprenticeship to gain one aspect of a skill before moving on to the next step (Kolodny, 1991). Cultural prohibitions permeate some societies; for example, values that discourage assertiveness, outspokenness, and competitiveness in some cultures result in behavior that can be interpreted as being indifferent,
having nothing to say, or being unable to act decisively (Hoy, 1993) (NRC, 1997b, p. 59). The problems engendered by these cultural differences are often beyond the ability of teachers of advanced courses to handle on their own. In many such cases, support from other members of the school staff is essential.
Finally, we note that advanced programs should use the resources of the community to maximum advantage. In many cases, local corporations are willing to make substantial contributions of equipment for the use of advanced physics programs. They also often run special programs and competitions that are highly stimulating to advanced students. Government laboratories and colleges can have a large positive impact on advanced programs as well by giving students the opportunity to meet practicing scientists and explore laboratories in which research at the forefront of physics is taking place. Teachers of advanced courses should avail themselves of these special opportunities in their communities.
ALTERNATIVE CURRICULAR MATERIALS FOR ADVANCED PHYSICS INSTRUCTION
Many excellent advanced physics courses can be developed using traditional physics texts because a skilled teacher can modify the material to incorporate interactive engagement techniques (Hake, 1998) that enhance student learning. It would be better, however, to have curricular materials that themselves directly support interactive engagement. This section describes several examples of such materials to demonstrate the variety of alternative resources available and to provide some concrete examples of what is meant by interactive engagement. The panel makes no claim of completeness in our listing, nor are we endorsing any particular set of materials.24
Most of the following materials have been developed and studied by researchers in physics education. In several cases, there is a significant body of evidence of improvements in student learning:25
Real Time Physics (Sokoloff, Laws, and Thornton, 1994, 1997) is a guide to microcomputer-based laboratory work designed to enhance students’ conceptual understanding of introductory physics topics. Studies of students using these materials have shown marked improvements in their understanding of concepts of kinematics and dynamics (Thornton and Sokoloff, 1990, 1998).
“Physics Tutorials” (McDermott and Shaffer, 2002) are a combination of written exercises and simple experiments that can enhance student understanding of introductory physics topics. Designed primarily for use in introductory college courses to supplement conventional curricula, these tutorials would be appropriate and valuable for use in advanced secondary courses. The most complete set of such
More information about the materials, along with links to the publishers, can be found at the Web site of the American Association of Physics Teachers’ Physical Sciences Resource Center, www.psrc-online.org. Many of the publishers maintain Web sites of resource materials that may be useful to teachers.
This evidence is not specific to advanced high school physics courses. However, it is reasonable to expect that results from introductory college-level courses should generally apply.
materials is that of the University of Washington Physics Education Group (1999) (McDermott, Shaffer, and Somers, 1994; Shaffer and McDermott, 1992; and Wosilait, Heron, Shaffer, and McDermott, 1999).
A Modeling Method for High-School Physics Instruction (Wells, Hestenes, and Swackhamer, 1995) focuses on having students actively engaged in defining, building, understanding, and testing mathematical models of physical phenomena. Students compare their models with both traditional models and the results of experiments. The method focuses on model building, testing, and deployment (using the model in new situations) as a key scientific activity (Halloun, 1996).
The Workshop Physics Activity Guide (Laws, 1997) is used as the main text for a 1-year calculus-based physics course (a traditional physics text can be used as a reference). The course features the use of integrated computer tools that students, working in small groups, use to collect, display, and analyze data and to develop mathematical models describing physical phenomena. Based on the results of physics education research, the guide uses a four-part learning cycle to have students (1) predict the results of an experiment, (2) reflect on the actual results and refine their predictions, (3) develop mathematical models with appropriate equations to describe the experiment, and (4) perform experiments to test the models (Laws, 1989, 1991, 1999).
Electric and Magnetic Interactions (Chabay and Sherwood, 1999) is a calculus-based introductory physics course that emphasizes the development of conceptual models and scientific explanations of electricity and magnetism phenomena. Kits of scotch tape, batteries, light bulbs, one-farad capacitors, and so on are used for “desk-top experiments” that introduce phenomena before students build detailed mathematical and conceptual models. The authors are just completing a text, Matter and Interactions (2002), to be used in the first semester of an introductory physics course. That text also emphasizes conceptual models and scientific explanations reinforced by computational studies of mechanical systems.
Six Ideas That Shaped Physics (Moore, 1998) is a calculus-based course that organizes physics topics around six big ideas. Each chapter includes many conceptual questions useful for in-class engagement, as well as a good collection of physics problems that require students to synthesize topics. The latter are particularly useful for collaborative learning groups. The book includes a good introduction to relativity, a section on quantum mechanics, and a section on thermal physics emphasizing a statistical mechanics approach. Early versions of the text were evaluated through the Introductory University Physics Program.26
Physics: A Contemporary Perspective (Knight, 1997) claims to be the first complete calculus-based introductory physics textbook reflecting the results of physics education research. It emphasizes a balance of qualitative and quantitative reasoning, multiple representations of knowledge, and a systematic approach to problem solving.
A companion student workbook helps students develop conceptual and graphical understanding.
The CASTLE curriculum materials developed by Pasco Scientific include material for investigating electricity, including batteries, light bulbs, compass needles, and large capacitance capacitors. The focus is on developing conceptual models of electrical effects, particularly in circuits.27
C3P: Comprehensive Conceptual Curriculum for Physics is a high school curriculum that draws on materials from a variety of sources, including videos, inquiry methods, and laboratory experiments.28
Although not traditionally part of advanced physics courses in high schools, quantum physics may well be used as an optional or enrichment topic. A set of materials called Visual Quantum Mechanics has been developed by Dean Zollman and colleagues in the Kansas State physics education research group. The materials include simple experiments that illustrate basic quantum physics phenomena using light-emitting diodes, glow sticks, and other inexpensive equipment. Computer programs help students build conceptual models to explain the results of the experiments.
Several publishers are revising traditional textbooks to take into account the results of physics education research. These revised texts emphasize student engagement in developing conceptual understanding, as well as improved approaches to problem solving. Already published is Physics for Scientists and Engineers (Serway and Beichner, 2000). In addition, a team of physics education researchers is producing a new version of Fundamentals of Physics (Halliday, Resnick, and Walker, 2000). This version is still in progress and should not be confused with the sixth edition of the textbook released in July 2000.
Some other resources that might be used in an advanced high school physics course include the following:
PRISMS (Physics Resources and Instructional Strategies for Motivating Students)—The primary materials include a teacher’s guide with 120 activities for use with introductory physics students.29
PTRA Workshops and Manuals—The Physics Teacher Resource Agents program (sponsored by the American Association of Physics Teachers and the National Science Foundation) has developed a series of workshops and teacher resources for most of the topics taught in high school physics. The majority involve hands-on projects for students, as well as other teaching materials.30
String and Sticky Tape Experiments (Edge, 1987).
Conceptual Physics (Hewitt, 1999).
More information can be found on the Web at http://www.pasco.com/ [4/17/2002].
For further information, see the Web site http://phys.udallas.edu/ [4/17/2002].
Further information is available at http://www.prisms.uni.edu/ [4/17/2002].
Further information can be found at the American Association of Physics Teachers Web site by going to http://www.aapt.org/ [4/17/2002] and clicking on Programs and then PTRA.
Active Physics (Eisenkraft, 1999)—This text is designed to be used as a “physics first” course, where students in ninth or tenth grade take physics before chemistry or biology. All of the units focus on applications of physics in everyday life, such as automobiles, home heating, and sound reproduction, with many hands-on activities.
Changing Emphases in Physics and Their Impact on Advanced Physics Instruction
The Panel notes two changes in the practice of physics that should eventually be reflected in instruction in advanced physics courses in high school: the increasing use of computers in a variety of ways, and a growing emphasis on interdisciplinary connections.
INCREASING USE OF COMPUTERS
The availability of easy-to-use numerical computation tools, graphical analysis programs, symbolic manipulation programs (e.g., MathCad, MAPLE, Mathematica), computer-based data acquisition, simulation programs, and the like is leading to dramatic changes in the practice of research in physics. As a result, there is somewhat less emphasis on detailed analytic calculations in many areas of physics research and an explosion of work in such areas as nonlinear dynamics and the study of complex systems, in which numerical and graphical tools are absolutely essential because traditional analytic methods fail. The panel argues that these same tools have the potential to effect major changes in the nature of introductory physics instruction.
The beginnings of those changes can be seen in the widespread use of computer-based and calculator-based data acquisition in introductory physics laboratory work. Computer simulations are also beginning to be used as a way of allowing students to explore actively the predictions and range of validity of various models of the physical world. These tools help introductory physics students build a better conceptual understanding of physical phenomena by actively exploring models and their predictions. Finally, and most provocatively, researchers have begun to pursue the use of computer programming as a medium and language for the study of introductory physics. In a manner that reflects the growing role of computational modeling in physics research, students may benefit by using computer programming as a means to express, explore, and refine ideas in physics (diSessa, 2000; Sherin, diSessa, and Hammer, 1993; Wilensky and Resnick, 1999).
The panel sees little in the current AP and IB curricula that reflects these changes in the practice of physics. We note that although many AP and IB teachers do make use of computers
in a variety of ways, neither the AP nor the IB examinations assess the use of computer tools. Accordingly, the panel makes the following recommendations:
Advanced high school physics curricula should, over a period of time, evolve to include more use of computer tools in ways that are effective in physics instruction.
Physics education researchers and others should continue to study the use of computers in introductory physics instruction (including advanced high school courses) to identify those uses that are particularly effective in promoting the instructional goals identified elsewhere in this report.
Eventually, cyberspace and information technology (CIT) may completely change the way new physical theories are developed. Instead of analytical models, theorists will use computer programs to conceptualize physical systems that cannot be adequately represented by traditional analytical methods. Students need exposure to such computer-assisted conceptualization as early as possible in their education. Therefore, the panel recommends that an appropriate unit in computer-assisted conceptualization be developed as soon as possible and made available to teachers of advanced physics courses as a possible optional topic (see Chapter 2 for a discussion of optional topics).
Capability for Rapid Information Sharing Among Physics Teachers
The primary opportunity afforded by the ever-expanding Internet is its promise of a new method of communication by which groups focused on particular sets of issues can rapidly interact to develop innovations. Teachers of advanced high school physics courses could form such groups to share and enrich their pedagogical skills. The opportunity now exists to create a national electronic clearinghouse of information relating to secondary school physics instruction. Box 5-1 presents a short list of some Internet groups for physics teachers that are already active.
Box 5-1. Internet Groups for Physics Teachers
The following are some active Internet groups for physics teachers. The descriptions of these groups are taken from the article “Communities of On-Line Physics Educators” by MacIsaac (2000), also available on the Web at http://pc53.phy.nau.edu/danmac/aug99listsmanu: