Squaring the Circle: a Mathematically Rigorous Physics First
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Squaring the Circle: A Mathematically Rigorous Physics First Robert Goodman, Bergen County Technical High School, Teterboro, NJ Eugenia Etkina, Graduate School of Education, Rutgers University, New Brunswick, NJ or more than a hundred years there has chemistry and physics, and chemistry requires a been a debate about the sequence in which foundation of physics. The reverse is not the case. F high school science should be taught. The sequence biology-chemistry-physics has been Many advocates of “Physics First” support a more considered to better align physics and mathematics, as conceptual ninth-grade physics course followed by most students do not learn basic trigonometry until chemistry and then biology while supporters of 10th grade, and trigonometry has been considered mathematically rigorous physics favor a sequence necessary for high school physics. This required phys- of biology-chemistry-physics. This paper describes ics to be the 11th-grade science, leading to the tradi- a new approach that incorporates some of the best tional biology-chemistry-physics sequence. arguments from both sides. The main idea is to teach As biology has become increasingly analytical, a mathematically rigorous ninth-grade physics course the importance of flow became stronger and led based on algebra alone, avoiding trigonometry.1,2 By to an increasing number of “Physics First” experi- selecting topics from the AP Physics B curriculum ments.11,12 However, the lack of trigonometry knowl- that form a foundation for both chemistry and edge amongst ninth-grade students gave impetus to a biology, one can establish an efficient science conceptual, almost “nonmathematical” approach in sequence. The goal of this paper is to describe this Physics First courses.5,6 This approach diminished the new approach to Physics First and present data value of Physics First to those who consider physics to supporting its effectiveness. be a mathematical science.4,7,8,10-12 A Problem: Logical Flow or Solution: Algebra-Based Ninth-Grade Mathematical Physics? Physics The sequence in which the sciences are taught The conflict between logical flow and the lack of has been an area of contention for more than 100 trigonometry knowledge among ninth-grade students years, since the Committee of Ten established the can be resolved by teaching a ninth-grade mathemati- current sequence in 1892.3 Should the sequence of cally intensive physics course based on algebra alone, science be physics-chemistry-biology or biology- without trigonometry.1 This approach represents a chemistry-physics? This conflict arose because the major departure from “conceptual physics” but is former sequence supports the logical flow of the within the grasp of anyone with a basic understand- sciences while the latter aligns physics and its requisite ing of algebra. A careful analysis of physics curricula mathematics. shows that only about 10–15 % of the content re- The sequence of physics-chemistry-biology is quires trigonometry. Students can comprehend the logical,4-10 as biology requires a foundation of both rest of the material and solve challenging problems 222 DOI: 10.1119/1.2895672 THE PHYSICS TEACHER ◆ Vol. 46, April 2008 only using algebra. The argument that ninth-graders We use a social constructivist approach that strong- are weak in algebra or are taking it in parallel with ly encourages students to work together to build physics only strengthens the argument for algebra- understanding. While conceptual understanding is a based physics. Students can master algebra while they core principal, so is problem solving. Students con- are learning physics, using physics as a context. Of struct their understanding when they are challenged course this approach will lead to some omissions in to apply new concepts to increasingly complex, inter- content. This can be addressed by offering students esting problems; much of the time in class is devoted who choose to take it a second year of physics, AP B to this. Physics, which is closely linked to the first-year course. Formative assessment is ongoing and essential to In the ninth-grade course the first semester is de- the entire learning and teaching process. In fact, the voted to mechanics and the second to electricity and only student assessments that are not formative in magnetism, both of which are essential to chemistry nature are the midterm, final exam, and the AP exam. and biology. Concepts such as energy, force, electric Students may, and are encouraged to, retake new ver- charges, electric field, and atomic structure are all sions of all other assessments; only their highest scores prerequisite for understanding chemistry and then count. biology; the ninth-grade physics course lays that foun- All major tests use the AP format (half multiple dation. choice and half free response) and scoring approach. Students can then move on to the second-year Tests are curved using a modified version of the curve physics course—AP Physics B. In this course they used for the AP exam itself (where a score of about spend some time solving those problems in mechanics 65% equates to a 5, their “A”). We only include grades and electricity that require trigonometry, extending on test and laboratory work in the student’s course these two topics, and most of their time on additional grade: homework, participation, etc. play no direct topics. The ninth-grade course can stress depth of un- role. The result has been a very strong correlation be- derstanding rather than covering many topics since all tween course grades and AP scores. students who are interested enough to study further All students in the school take the same tests on the can take the AP B course. Developing problem-solv- same day. This extends the social constructivist ap- ing skills and conceptual understanding are two inter- proach beyond the classroom to the entire school. All connected goals of the course. students can work together to prepare for tests or re- While a key advantage of this approach is to build tests and to complete laboratory work. This has facili- mathematical skills in the context of a physics course, tated another key element in the program, establish- this does not require strong coordination between ing after-school sessions twice a week where students, mathematics and science teachers, nor would that co- in the same or different grades, and teachers work ordination be generally possible. This is because stu- together in a collaborative, constructivist fashion. dents will be in different math classes while taking this course (i.e., while most are in Algebra I, some are in Second Year Physics: The Completion Geometry and Algebra I, some are in Geometry alone, of AP Physics B and others are in Math Analysis I). All students in the school complete about half of the AP Physics B curriculum by the end of ninth Pedagogical Approach grade. They are well positioned to complete AP Phys- All students in the school, with a wide range of pri- ics, if they choose to take the second-year course, and or experience and success in mathematics and science, then take the exam (a requirement for all AP courses take a mathematically rigorous physics course in ninth in the school) in 10th grade or in a later year. They grade. Being successful with such a diverse population simply add the extra level of mathematical sophistica- required developing appropriate pedagogical tech- tion required for trigonometry-based problems to niques and curricular materials. Due to space limita- their ninth-grade study of mechanics and E&M and tions, a thorough discussion of these has been reserved the additional topics from the AP B curriculum. for an online appendix.13 However, space does permit This new way of dividing that AP B curriculum a general description. THE PHYSICS TEACHER ◆ Vol. 46, April 2008 223 biology course draws its content from the AP Biology TIMSS Results for Student Achievement in Physics 700 curriculum. By choosing the proper set of topics, we 586 581 600 573 created three science courses, in physics, biology, and 545 534 523 522 518 511 494 488 488 486 485 466 chemistry, that each serves as either the first half of an TIMSS 500 451 435 423 400 AP course or as a strong stand-alone course. We also oints on 300 take advantage of the overlap between the AP topics 200 in physics, chemistry, and biology to eliminate redun- 100 dancy and increase curriculum efficiency. Number of P 0 k c y e d. ay Is AP Emphasis Productive? Fe U.S. anc rw ustria yprus Latvia Fr C A Greece ustralia Canada No ysics 3+ Sweden ysics 1,2 Slovenia A Denmar German Switzerland Russian One might wonder if the emphasis on the AP ech Republi AP Ph AP AP Ph AP Cz physics, chemistry, or biology is a productive way Fig. 1. TIMSS Report: International Student Achievement in to improve science education. A strong support for Physics. our argument comes from the study of Gonzalez, over two years, to match against the growing math- O’Connor and Miles,14 who reported that although ematical skills of the students, is fundamental to this overall the United States performed at the bottom of new approach. A good example is how we approach the list of 16 countries in the 1998 TIMSS (Trends in the topic of projectile motion. Mathematics and Science Study) Physics test, students Traditionally, vectors appear early in a physics who took the AP Physics exam and scored less than course, immediately subsequent to the development a 3 were in the middle of the list. American students of kinematics. They can be oriented at any angle, re- who passed the AP exam, by earning a score of 3 or quiring knowledge of trigonometry.