Sequencing Science and Math in 7-12th GradeJohn D. Mays
Putting Physics First
We begin by considering where physics fits into the upper-school program—a key factor. On this issue, it is essential to keep two things in mind. The first is the distinction between an introductory physics course, which does not use vector computation methods, and a more advanced course that incorporates vector methods into problem solutions. In this paper, these will be referred to as the introductory course and the vector/trig course.
The second issue is the need for all students to have an appropriate physics course, that is, one that fits both their aptitude and their academic interests. All students should receive an introduction to physics in high school. However, for students of average ability (in math and science) to take physics, they need to take it at the introductory level. Vector/trig physics is very difficult and is not for everyone, despite current fashions in state standards requiring all students to take a vector/trig physics course. Such a requirement is far from reasonable. Administrators should carefully consider whether to follow this fashion off the cliff or adopt more sensible programming that serves the needs of all students.
Some schools offer even more advanced physics courses such as one or both of the College Board’s AP Physics courses. We will not consider this possibility here. Relatively few students are up for such advanced courses in high school and relatively few small schools can afford to incorporate such courses into their program. In fact, schools can offer a robust program that prepares students for college well, and place those students into select colleges, without the complications that come with AP Physics courses.
There are essentially three options for placing physics in the high school curriculum. The first is to offer only vector/trig physics to advanced upper level students (juniors or seniors). This programming model was common 40 years ago. The problem with this option is that only a small percentage of a typical student body can handle a course like this. When the vector/trig course is offered as an elective to upper level students, typically about 10–20% of the students elect to take it. This means some 80–90% of the students do not have an introductory physics course in high school at all, which probably means no physics course ever. For schools that desire to offer a robust science program this is not an acceptable scenario.
The second possibility is a result of new science standards in some states. This option is to require all students to take a somewhat mild vector/trig physics course during their junior year, while offering a more robust course (such as an AP course) as an elective in the senior year. As popular as this option is becoming, it nevertheless is not at all reasonable. As mentioned above, less than a quarter of a typical student body can handle a vector/trig course, and this applies even if the content is watered down. From our own classroom observations—which include large, prestigious independent schools (over 500 high school students)—we have concluded that this approach is an unmitigated disaster. For example, one case we encountered was at large private school with a reputation for rigor. The teacher was competent and the students had all the advantages that come with private education. On a chapter test administered in April, the class average was 63, and several students were overheard making pitiable jokes about their scores in the 40s. It was evident that the educational process had been reduced to farce. Despite the ill-advised pressure from some state standards boards, this option is really not an option. Vector/trig physics is not for everyone.
The third option is to teach introductory physics to all freshmen, and then offer vector/trig physics as an elective for upper level students (along with other upper-level science electives). This is a proven option that appropriately addresses the needs of virtually all students. Grade-level freshmen should take an introductory course that focuses on the basic principles of physics. The course should incorporate plenty of mathematics, but the math should be limited to what students in first-year Algebra can handle. The inclusion of mathematics in the course is very important, and for this reason we do not recommend offering so-called “conceptual physics” to these students. Even students with modest ability can solve basic problems in motion, force, energy, and density, and such computations are a key point of integration between science and math curricula. The introductory course for accelerated or honors-level students should incorporate introductory chemistry as well.
We have spoken of two different groups of freshmen: grade-level students who take introductory physics, and accelerated students who take introductory physics and chemistry. The assumption here is that a school’s program is stratified into two different pathways (aka, tracks): a grade-level pathway and an accelerated pathway. We will address this issue briefly in the next section.
So far we have seen that placing introductory physics in the freshman year provides all students with an introduction to the subject, while allowing for higher-aptitude students to take a more advanced vector/trig course later as an elective. This argument for course placement is based on the practical but paramount concerns of student aptitude and appropriate placement. But another justification for placing introductory physics in 9th grade is the significant benefit of having a background in physics prior to taking chemistry or biology. The introductory physics course curriculum includes a number of topics that are fundamental in the study of chemistry, including electric charge, energy, heat, energy transfer, phases of matter, electrostatic attraction, temperature scales, light, types of substances, and the internal structure of the atom. Studying these topics in introductory physics as freshmen pays large dividends when students encounter them later in chemistry.
A solid introductory physics course should also provide students with a significant amount of practice in basic scientific mathematical skills. Three skills of supreme importance in science are working with metric system units and prefixes, performing unit conversions, and using scientific notation. All students should completely master these skills as freshmen. Additionally, introductory physics should introduce students to the roles of accuracy and precision in scientific measurements, and give students a lot of practice working with significant digits.
With a background in these essential skills, students are much better prepared to tackle topics in chemistry. Under the conventional biology–chemistry–physics sequence, chemistry is often perceived as quite difficult. One reason for this is that students typically have to learn chemical principles and mathematical skills simultaneously. However, when students arrive in chemistry having already mastered unit conversions, metric prefixes, scientific notation, and significant digits, a lot of the perceived difficulty of chemistry disappears.
In summary, placing introductory physics in 9th grade for all students provides the best access for students to have an appropriate introduction to physics. It also introduces all students to important fundamental topics that play major roles in chemistry. Finally, studying physics in the freshman year gives all students an opportunity to master critical mathematical skills. With these skills in their toolbox prior to taking chemistry, the road is clear to tackle the basic topics in chemistry without being distracted by the need to learn essential math skills at the same time.
Dual Science and Math Pathways
We strongly advocate providing at least two pathways for high school mathematics. It is a fact that on average roughly half the students of average or higher ability are ready for Algebra in 8th grade. The other half need an additional year of work with pre-algebra before taking Algebra in 9th grade
Math placement is a complex topic, and we have written at greater length in another paper on the reasons for stratifying math students into at least two different pathways (see our white paper, “Stratifying Math Students”). Suffice it here to say that stratification is essential, and schools should place a high priority on providing separate pathways for grade-level students who take Algebra in 9th grade and accelerated students who take Algebra in 8th grade. (One or two percent of students are ready for algebra in 7th grade. If necessary, these students can be placed with the accelerated 8th graders.)
When students are stratified into two pathways in math, the same stratification should generally apply as well to the high school science courses. Splitting students into two science pathways allows the school to provide a solid, basic sequence of courses for grade-level students, while enabling accelerated students to undertake a challenging curriculum of science courses that positions them to compete for admission into technical majors at more selective colleges.
Science and Math Linkage
Figure 1 shows the dual-pathway science course sequence we recommend. There is a key linkage connecting students’ math placement and the science courses they undertake each year. This key linkage relates to the math prerequisites for studying chemistry, and is one of the important factors influencing science course sequencing in both grade-level and accelerated pathways.
The prerequisite in question is the need for students to take (or have completed) Algebra 2 at the same time they study chemistry. Assuming a standard sequence in math courses (Algebra, Geometry, Algebra 2), grade-level students take Algebra 2 as juniors and accelerated students take it as sophomores.
A number of Algebra 2 topics come up naturally in the study of chemistry. The definitions of pH and pOH are logarithmic expressions, and solving pH problems involves both logarithms and exponential functions. Reaction rates and chemical equilibrium involve power functions. The inclusion of these topics in chemistry requires that students take chemistry concurrently with (or after) their second year of algebra. This, in turn, places chemistry in either the sophomore or junior year, depending on math placement. The placement of chemistry into one of two different years allows for some key distinctions in the two science course sequences that allow the school to serve each group of students appropriately. We will look at these in the next section.
The Complete Science–Math Course Sequence
Putting the science and math courses together for both pathways results in the program shown in Figure 2. Two or three points should be noted right away. First, Anatomy and Physiology is a good fit for grade-level students in 12th grade. The topics covered align closely with their natural interests, and the course is good preparation for college study.
Second, the math course shown in the chart for grade-level seniors is AP Statistics. The alignment between the AP Statistics syllabus and what would be taught in a stats course anyway is nearly 100%. Setting up the course with the College Board as an AP course gives grade-level students a highlight on their transcripts. We have seen AP statistics taught even to students with very limited math ability and have found it accessible to all but special-needs students. Further, statistics is useful preparation for nearly every college major. We are less enthusiastic about Precalculus as a senior course for grade-level students. We have found that the material is unnecessarily challenging for them and the abstract topics seem irrelevant to many students.
Third, when student numbers permit, it is useful to stratify the junior and senior math offerings for accelerated students one step further. Students with appropriate aptitude and interest can be placed in a “pre-AP” precalculus course, followed by AP Calculus. Bright students who aren’t quite up for the extreme rigors of AP Calculus should take a separate precalculus class followed by a non-AP calculus course. Just as with vector/trig physics, there will be many students—even those in the accelerated pathway—who cannot handle AP Calculus. A non-AP course moving at a slower pace is right for them.
The considerations we have addressed so far have justified the grade-level science pathway. We haven’t mentioned the grade-level sophomore biology course yet, but clearly the considerations for 9th, 11th, and 12th grade leave us with an opening in 10th grade that is appropriately filled with a general biology course.
Some other important factors are involved in the accelerated science sequence. To be competitive at highly selective colleges and universities, the accelerated students need a heavy dose of cell chemistry in their biology course. This is much easier to accomplish when chemistry precedes biology. Happily, the math alignment in the accelerated pathway facilitates this sequencing.
Now observe the line-up we have in the accelerated science course sequence. The introductory physics/chemistry course covers the physics material by mid- to late February and then switches to chemistry. In the final three months of the year, the students get a solid head start in chemistry, allowing us to denote the sophomore chemistry course as Advanced Chemistry. The introductory chemistry material their freshman year allows them to move quickly into more advanced topics their sophomore year (particularly if the freshman course is taught with mastery and retention in mind). With chemistry under their belts, the students’ junior biology course can include a full semester of cell chemistry, along with other standard topics such as Mendelian genetics. To make room in this course for a full semester of cell chemistry, less sophisticated topics such as human organ systems should be moved down to the middle school life science course, where they are quite age appropriate. After taking advanced biology as juniors, students are ready for a solid course in molecular biology their senior year, a course that is always impressive to colleges. Alternatively, students can take the vector/trig physics class. Schedules permitting, science-minded students typically want to take both.
Considerations for Middle School
We conclude with a few considerations for middle school science. The best courses to offer are Life Science, Physical Science, and Earth Science. An astronomy component in the Earth Science course is a good idea, and makes the course a lot of fun. We do not recommend omnibus such as “general science.” Such courses tend to be amorphous and lacking in focus. Middle school students learn more and remember more if they are able to focus on one basic discipline for the entire year.
If your school includes 6th grade as part of the middle school, then we recommend Life Science–Physical Science–Earth Science as the sequence for 6th, 7th, and 8th grades respectively. If the middle school consists of only 7th and 8th grades, then you must choose two out of these three courses to offer in the middle school. For schools incorporating both elementary and middle school grades, one solution is to schedule physical science and earth science for 7th and 8th grades, and make life science the key topic for 6th grade, even though the 6th grade is part of the elementary school. Another approach is to switch the physical science and life science years and still keep one of them in 6th grade. The only sequence we would not recommend is placing physical science in 8th grade when introductory physics occurs in 9th grade. There is a lot of overlap between these courses, and it is best to spread them apart by at least one year.