The only thing I’d add is partial fractions. I find it very disturbing to teach partial fractions in the middle of integration. Everybody who starts calculus should know how to do partial fractions.
To address questions about the difference between courses sequences in high school versus in college:
In my high school, the courses were Algebra 1, Geometry, Algebra 2, and Pre-calculus. Trigonometry was covered in Geometry, and Algebra 2; there was no course in my high school named Trigonometry.
At the college I attended, students were expected to know the subject matter in Algebra 1. There was a remedial, non-credit workshop which covered (basically) Algebra 1.
From there, a student could either take Math 3 and Math 4 (which were named College Algebra (basically Algebra 2) and Trigonometry, respectively) or Math 14 (named Pre-calculus) before beginning the series of calculus courses. Math 14 was for students who had taken sufficient math classes in high school; it covered the same subject matter as College Algebra and Trigonometry, but in only one semester instead of two. Students who hadn’t taken as much math in high school were expected to take both one-semester courses College Algebra and Trigonometry.
I guess my long-winded point is that the topics covered in a Pre-Calculus class would depend on whether it’s in high school or college. The high school version might cover more things over a longer time.
Ex Teacher here, a little high school and some college.
Ahhhh, Pre-Calculus. The course schools need, but in the rural areas I was in, was seldom taught (at least long ago when I did this).
Schools love the idea of being able to say ‘Our students take Calculus’. The problem is, their students don’t take Precalculus. Taking Calculus in high school is great ONLY IF they also take Precalculus. So they need to push things back a year. Algebra 1 in 8th grade etc. However, these rural schools couldn’t do that…but they want to say "Our students take Calculus’. I was trying to follow what my Math profs said which was to teach Precalculus instead of Calculus but wasn’t allowed.
I really was able to see what they meant when I taught college. In Calc I, students that had Precalculus AND NO CALCULUS VASTLY outperformed students who had Calculus in High School but no Precalculus. It was amazing.
Precalculus was mainly for 2-3 things. To really nail down their Algebra skills, introduce Analytical Geometry and to teach Trigonometry. Those subjects vastly help students going into college more than getting introduced to Calculus.
Interesting! Thanks.
I think it’s increasingly clear that, absent any other information, “Precalculus” by itself does not always mean the same thing. Also, the mathematics curriculum in high school for a future mathematics student is never going to be the same as for a classical tripos.
There are more specific terminology and standardized curricula available. For instance, a high school student in an IB diploma program who wants to study business management or chemistry would study algebra, functions and equations, trigonometry, vector algebra, statistics and probability, and calculus as part of the Mathematics SL course; a prospective engineering student would study all of that in much greater depth as well as at least one optional topic of further statistics and probability, set theory, series and differential equations, or discrete mathematics. While students who do not expect to need much mathematics (social scientists and economists? I feel this tracking is a bit unfair) have their own Mathematical Studies SL curriculum.
That’s one thing I distinctly remember learning in my own high school Precalculus class. But many students don’t see it until they get to Techniques of Integration in Calculus II, and I’ve never taught it in any college class below that level.
The thing is, I can’t think of any place it’s used before then (though there may well be one). So I’m fine with students not learning it until they’ll actually use it. I don’t find it “dsiturbing” to teach in the middle of integration; it’s just part of the topics covered there.
My SIL teaches a precalculus course at a commercial test prep company and also tutors it to local high schoolers & college kids.
Precalc, at least here in FL, today is very different from what the term meant 40+ years ago when most of us took it. She’s talked about how “weird” it feels compared to what she taught under the same label in high school just 10 years ago. I’ve not managed to absorb the details of which is which. Next time I see her I’ll be sure to capture the differences.
I am not surprised. This kind of thing is part of the motivation behind the Joint Position Statement of the Mathematical Association of America and the National Council of Teachers of Mathematics on Teaching Calculus that was released a few years ago.
Both social scientists and economists would benefit from a little more math than they often get, but not primarily pre-engineering math or pre-mathematics math.
Social scientists often have to study basic statistics. Economists would benefit from pre-calculus algebra including limits, series and areas – but since they seldom have it, anything requiring it may be left to optional advanced units.
Calculus certainly can be used by economists.
It’s fairly typical for colleges and universities to offer a “Business Calculus” or “Calculus for Social Science” class as an alternative to the standard Calculus sequence that STEM majors take. One of the big differences is that such a Calculus class makes no use of trig functions and thus does not require any trigonometry as a prerequisite.
Thudlow, Nice to see my perceptions might actually have been real 
I have to confess, I did cheat. When forced to teach Calculus in high school, I was allowed to choose my book. I chose a book that said Calculus on the front but over half of the first part of the book was devoted to Precalc, especially Trig. I just made sure to keep my mouth shut.
In fact, economics uses calculus more than stats, I’d say: economics is all curves and functions.
Taking another look at that IB curriculum I mentioned (chosen because it is a somewhat well-known standardised curriculum, not because I endorse it or anti-endorse it), the standard levels for “students who do not anticipate a need for mathematics in their future studies”
The IB DP mathematical studies standard level (SL) course focuses on important interconnected mathematical topics. The syllabus focuses on: placing more emphasis on student understanding of fundamental concepts than on symbolic manipulation and complex manipulative skills; giving greater emphasis to developing students’ mathematical reasoning rather than performing routine operations; solving mathematical problems embedded in a wide range of contexts; using the calculator effectively. There is an emphasis on applications of mathematics and statistical techniques. It is designed to offer students with varied mathematical backgrounds and abilities the opportunity to learn important concepts and techniques and to gain an understanding of a wide variety of mathematical topics, preparing them to solve problems in a variety of settings, develop more sophisticated mathematical reasoning and enhance their critical thinking.
NB it does include both trigonometry and calculus. Just not from the same point of view, depth, or emphasis that the students would get from a more technical track:
- Introduction to the graphic display calculator
- Number and algebra
- Sets, logic and probability
- Functions
- Geometry and trigonometry
- Statistics
- Introductory differential calculus
- Financial mathematics
I used Saxon’s Advanced Math. I found a very small table of contents on this website:
https://www.christianbook.com/saxon-advanced-math-home-study-kit/9781565771277/pd/791273
I don’t even remember what half this stuff means.
My son took Geometry in 8th grade and just finished Advanced Algebra for his first trimester as a freshman. He just started precalc which runs two trimesters.
According to the teacher they will cover this trimester: The Nature of Graphs, Conics, the Trigonometric Functions, and Graphs of Trigonometric Functions.