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Stewart E. Brekke
The language of physics is primarily mathematics. To teach a physics course to students without mathematics is not only deluding him/her, but also cheating that student out of the understanding of the true nature of the physical universe. Most teachers would not hurt their students in that manner–i.e., by teaching them something that is not true or half true. One reason physics first or physics last in high school must be mathematical is that the mathematical course enhances logical thinking and reasoning which transfers to all aspects of life. This type of thinking helps all kinds of students to deal with future academic work, but also with problem solving situations in their lives. The qualitative course does not have the power to develop rationality the way in which the mathematically based course does. Further many students are mentally disorganized and the mathematical course helps them to organize their thoughts. Literacy in science, especially in physics and chemistry, is not gained by memorizing definitions and concepts. Rather, it is the capability of doing physics with mathematically based problem solving, manipulating formulas, taking data and analyzing it quantitatively. One criterion for measuring physics literacy in a student is determining if that student can calculate the average speed of an object given the distance traveled and time elapsed. The language of physics is mathematics, and if we do not give the student the mathematical course, we are deluding that student with misinformation, making the student think that he knows a subject, physics, when he really does not. Certainly, knowing concepts and ideas of physics is essential, but also doing the mathematical portion, the main portion of physics, is a fundamental aspect of any physics course, physics first or physics last.
In my experience in teaching physics and chemistry to ‘at risk’ students in the inner city schools of Chicago, I have found that there are many students in the average high school, majority and minority, capable of doing mathematical problem solving high school physics, not just those in the upper 30%, provided the physics teacher makes an above average effort to teach scientific notation, units, and the single concept approach per class session, with drills and practices, including the algebra and trigonometry when needed. With the mathematical physics course, first or last, the lives of our high school physics will be enhanced. We cannot change their lives with a qualitative course, such as conceptual physics, which does not really provide a solid foundation for learning any science, such as chemistry or biology, nor prepare them for any worthwhile career, nor rational thinking in an increasingly technological world.
We therefore cannot leave an optional mathematical physics course for the senior year of high school. If the student takes the non-mathematical Physics First course and later does not take the mathematical algebra-trigonometry physics in his/her last years, he will probably be locked into a lower paying career since he will not readily master the standard mathematically based college chemistry and physics courses upon which careers in medical fields, engineering and the sciences are based. That student will probably have a lower paying job the rest of his/her life in a non-satisfying career. This is especially true for minority students who are often given non-mathematical physics and/or chemistry in high school depriving them of an opportunity to improve their economic status. A number of these minority and majority students try to take a real chemistry and/or physics course, mathematically based, in higher education only to find themselves lost and far behind the other majority and minority students who have had the mathematically based physics course in high school. Often, these intelligent majority and minority students have the mental ability to do a standard mathematically based physics and/or chemistry course, but do not have the problem solving skills to readily master a quantitative type physics and/or chemistry course, thereby thwarting their upward mobility in society through good paying scientific, technical or medical careers.
Placing physics first in the high school science curriculum is supposedly done to provide a solid scientific foundation for the understanding of high school chemistry and biology, since physics is the subject upon which all sciences are based. Often, in a course called “conceptual physics”, given often as the Physics First course, very little or no mathematics is used. One real reason conceptual physics does not utilize much mathematics is that most high school students in the United States take basic algebra, the mathematics needed for the standard basic high school physics course, in the first year, usually at the same time they are taking Physics First, thus depriving the ninth grade physics student, it is often thought, of sufficient mathematical background to do the standard basic high school course. Also, many students, from above average to average, especially in the inner cities of our country, are weak in arithmetic, fractions, decimals and long division, and for many years were given non-mathematical science courses. Because of this weakness in arithmetic and algebra and lack of insight, many high school physics instructors simply gave up on these otherwise bright and capable students and offered them a non-mathematical physics or physical science course.
Even Paul Hewitt, a founder of the current “conceptual physics” approach, has recognized the need for more mathematics in the conceptual physics course. He has written a problem solving manual to accompany his text “Conceptual Physics” and has pointed out that in his opinion we need “both” the quantitative and qualitative aspects of the introductory physics course.
Physics teachers in high school and higher education should realize that for many years basic algebra and trigonometry have been taught in the elementary school in 7th and 8th grade. Also, I have found that taking a period to teach or recall some basic trigonometry with high school physics students, or teaching the basic trigonometry as the physics course progresses to ‘at risk’ students, is entirely feasible and pays dividends in higher level physics learning through mathematical problem solving.
In teaching at risk inner city students, average to honors, I have found that many students, even learning disabled students with math problems, can overcome the arithmetic barrier using basic inexpensive pocket calculators thereby virtually eliminating the fraction, decimal and division problems encountered in a basic physics or chemistry course. Furthermore, even if the student has not had formal algebra in elementary school, basic use of formulas is done throughout the elementary student’s mathematical education, such as using the formulas for the areas of various geometrical figures, like squares and triangles. Even the most at risk students can substitute a value for a variable with a little help from the instructor and then use his pocket calculator to produce a correct answer. Especially if the physics teacher, concentrating on one problem type, not only puts an example problem on the board showing the students how to substitute, but also if he goes around the room helping individual students to use their formulas, with drills and practices, substitution behavior will be mastered by many students. My experience has shown that if this learning method, using a single problem type, and going around the room helping individual students, with drills and practices on that single problem type, is done in the first part of the mathematical physics course, most students, average to honors to ‘at risk’, will become mathematically literate physics problem solvers. This approach makes it easier for the high school physics teacher to reach all the students in his classes, not the just the few that can afford tutors or have a college educated relative who can show the average student how to solve his physics problems. We must remember that the great physicists such Bohr, Planck and Rydberg, all had tutors to help them pass their physics exams and solve their physics problems.
The Physics First teacher, in my opinion, must teach the basic algebra and trigonometry himself as the Physics First course progresses through the year, and not wait for the students’ algebra teacher. This takes valuable physics time, especially at the beginning of the school year, but pays great dividends as the physics course continues through the year. In reality physics time is not very valuable unless it has a solid mathematical foundation. Furthermore, many more students will take ninth grade physics first and later skip a final mathematical physics course in his/her senior year.
In conclusion if we do not provide a solidly mathematical Physics First course, we are not only depriving the student of a true experience in physics, but also giving that student a misconception of what physical sciences are, possibly setting the students up for failure or disaster when they take a mathematically based physics or chemistry course in higher education, let alone limiting otherwise capable and intelligent students to lower level careers, thereby destroying their upward mobility and chance of equality in an increasingly technological society.
Stewart E. Brekke is a retired high school physics and chemistry teacher from the Chicago Public Schools, and lives in Downers Grove, IL