In theory, “college level” math starts at calculus, since high school college prep math theoretically goes up to precalculus. However, in reality, the levels of math courses taught in colleges substantially overlap with those taught in high schools, since the quality of high school math preparation that entering college students have varies a lot. Even elite non-STEM-focused schools offer precalculus courses (e.g. Princeton MAT 100) or calculus with review of precalculus courses (e.g. Harvard Ma 1a, 1b). Open admission community colleges may start with elementary algebra (= high school algebra 1) because some students need that.
@VMT @frazzled2thecore - I haven’t read my PM yet, sorry, and I will, but - regarding Diff Eq before college, my question would be this: suppose a kid has taken BC Calc and whatever other math their school offers. At that point, I’m not 100% sure that the next course needs to be or should be in the Calc sequence (Multivariable or Diff Eq). It’s true that my own child chose something else, but even apart from that, I think there is some breadth to be gotten as well, like maybe Topology, other Analysis courses, Probability/Statistics beyond the AP Stat curriculum, Data Structures, etc.
My philosophy of high school, whatever the level, is that exploration and breadth are also really important. (I think that can include any class, and it isn’t always the Alg-Geo-Calc pathway.) As a somewhat mathy person myself, I love the idea that a smart high-schooler can be thinking about other kinds of math in addition to learning more techniques to apply if/when s/he is an engineering major.
One thing about high school is that kids often take courses that they’ll never pursue farther again (e.g. AP history classes for a kid who will be an engineer). I.e. HS can/should be an opportunity to stretch your wings a little bit off the path. I say this as someone who LOVED Diff Eq, by the way. 
“@frazzled2thecore I believe a very small percentage of students enter college having taken differential equations.”
This is sort of a duh, because the average high school in America doesn’t have differential equations on the radar screen anywhere. Good grief, not every high school in America is super-affluent striver suburbia!
I’m a mathy person too, majored in mathematics (I too loved diff eq!) and still, I think there is plenty of time in college that no one need go above Calculus BC in high school. I dislike this race-to-get-it-all-in, as though it’s a race to finish everything prior to graduating hs and a race again to pack it all into 4 years. One CAN be a lifelong learner, you know. It really matters not one whit your first job out of college whether you learned diff-eq or anything else in high school or college.
In college, students who complete calculus 2 or come in with calculus BC credit often have several choices, and may need zero, some, or all of them for their majors: multivariable calculus, linear algebra, differential equations, discrete math (linear algebra and differential equations are sometimes combined into one course).
Where a post-calculus-BC course is offered in high schools, the most common one is multivariable calculus, although a few high schools with a significant population of super-advanced math students may offer more.
There is no reason to push a student onto a track that leads past calculus BC in high school. However, there is a very small number of students whose interest and ability in math will lead them that way even if not externally pushed, so they should not be held back if it is possible to accommodate them.
When I was in high school, the good students were one year ahead in math (which meant calculus BC instead of precalculus in 12th grade). There was no pushing beyond that, although an occasional student every few years would be two years ahead (calculus BC in 11th grade) and known as the top student in math. It does seem like there is a lot more inappropriate pushing ahead going on these days.
@TheGFG : The Larson textbook you referred to is a standard applied Calc text for business majors at many universities. I would agree with @fretfulmother that it may not be be the most interesting text to expose the beauty of calculus. The problems in there are fairly standard , follow the example types.
However, the teacher is assigning outside work that probably has little relation to the text. Sounds like fodder for a healthy tutoring business. That’s what’s going on in my kid’s high school in math and physics, even in the non-honors classes.
@ucbalumnus - thank you, yes, I know about the math series in college. My questioning was along the lines of, is it really necessary to continue The Calculus Path rather than getting some breadth, for the student who has more high school time left to devote to mathematics and has successfully completed BC Calc. It’s more of a theoretical academic question, I think.
Re #886
As a practical matter, post calculus BC math in high school options are likely to be limited by such factors as what may be offered at the high school and nearby colleges (where scheduling can work and cost is not a limitation).
gfg the class your daughter is taking can be made very difficult by the teacher. It just sounds like a bad situation. I feel for you and your daughter. It seems like all school districts are different.
@VMT - If the typical freshman is entering with AB calc, then it is likely that the school has adjusted its instruction to this level. And if most students are getting through the sequence, including those who enter with pre-calc, then the college is likely providing good instruction and good supports, and is invested in retaining students in STEM majors.
@fretfulmother - I agree with you about high school opportunities to acquire breadth and enrichment. And, frazzled kids have told me that they too LOVED diff eq.
I don’t know about individual schools, but it seems to me that at some schools (elite schools, highly-ranked state schools) many students are beginning engineering and physical science majors with either a high school BC class that has covered a good bit of multivariable calculus or a high school multivariable class, and are not moving into diff eq as their first class.
This can become a problem if instruction and evaluation in the calc 3 class is geared towards these students, and students without this background are not given a heads up. Some colleges adjust by making supplementary supports such as extra tutoring freely available to students, and alerting incoming students that their knowledge base is lagging behind classmates. Or, they might encourage students not quite ready to move to the next level to take an honors version of calc 3.
I agree, fewer even at elite schools are taking diff eq in high school, just as few pre-meds have been exposed to organic chemistry before entering college, and it seems less likely that extensive prior exposure will give students an advantage in these classes. But, that could change.
A wide range in the knowledge base among students entering a college classroom is not an issue limited to math classes, btw.
.
The Larson book is a text for students who don’t have a strong grasp of algebra or trigonometry, but who need to learn some calculus recipes. It has long, loving descriptions of what a Cartesian plane is, what the order of algebraic operations are, and other basic topics. Every single chapter has a section labelled “Algebra Tutor” that reviews some basic algebra. These are high school math topics. Students for whom this is the right text either never studied pre-calculus, or forgot much they learned there. But high school juniors with a weak grasp of algebra and trig don’t belong in calculus.
Every high school has classrooms full of juniors who don’t have a strong grasp of algebra and trig. That’s not a surprise, and there’s no shame in it. The folly is taking a classroom full of juniors who are weak in algebra and trig, and teaching them calculus recipes, instead of putting them in pre-calculus so they can get a good high school mathematics foundation.
If these students are stuffed with rote rules on how to differentiate a sine function and how to integrate a polynomial, they will forget the rules in a few months. They will have gained nothing, and they will still be weak in algebra. Either they will go to college and attempt calculus, but without an adequate mathematical foundation, or they won’t study math in college and will go through their lives with a weak grasp of what they should have been taught in high school.
High school is supposed to give students a solid foundation in algebra. That’s beneficial for most people, whether they will go further in math or not. This school system is rushing students through high school math and into calculus so that the school system and parents can boast about how advanced the student body is. They’re doing their students a disservice.
I agree but parents need to understand how very small this number truly is. I have a nephew who took Calc BC as a high school freshman. He was very very unusual even among his classmates at one of those top competitive test-in high schools. I saw parents who didn’t understand what it meant to be truly and highly gifted in math so they did push thinking that if their child was on track to take Calc BC as a junior (starting with Algebra I in 7th grade), then the child would somehow be (become?) a math genius.
What TheGFG describes is this situation going amok to the point where kids who should never have been in that track are “trapped” and suffering. I don’t know how much of this is driven by parents (“my child won’t get into even Rutgers”) and how much is driven by the school district (“we need to raise the number of students taking AP Calc classes”).
I know it was tough going against the grain. I think my child came out of it better but it was tough until she found her footing in the social/academic hierarchy at her high school (the top kids realized she was no dummy when she appeared in their classes). Often times, I was made to feel I focused on the wrong thing - thorough understanding of whatever subject (I know math, so that’s my example) - that strong solid foundation @“Cardinal Fang” describes and was unreasonably holding my child back.
Agree. When I was in high school, there was maybe one such student every few years who completed calculus BC before 12th grade (each high school class was about 400 students or so). About a third of graduates at the time went to four year colleges, and about a fourth of those were one year advanced in math (calculus BC in 12th grade).
What @TheGFG describes with four levels of high school calculus is completely ridiculous.
"As a practical matter, post calculus BC math in high school options are likely to be limited by such factors as what may be offered at the high school and nearby colleges (where scheduling can work and cost is not a limitation). "
@ucbalumnus - sure, but we are very fortunate in our district. My son had several math teachers in the department willing to do an independent study with him if he just picked the topic. I’m aware enough to realize that we are lucky enough to have:
- trained/smart/willing math teachers (they do this without compensation so it’s totally up to them if they want to)
- a child who not only has a penchant for math, but has the kind of personality to carry off an independent study
- a schedule in school that allows both the student and the teacher to have corresponding prep blocks
- my DH and me being aware enough of options and having the time/resources to find a nice text to use
Many kids, of course, do not have these privileges!
By the way, there is no specific acknowledgement of DS’s Independent Studies on his transcript, though they are official through the school. He wrote a sentence or two about each on the “transcript addendum” and they get a “Pass” (or I suppose, it could have been “Fail” 
if applicable) but no grade or weighting. It was up to DS and his LoR to make sure the content was represented. I think this system lends itself more to kids who want to learn for the joy of it, and less to kids who would want to rush through the system in some way.
I’m surprised by how many posters here liked differential equations as it was taught when we parents were college students. Mr. Fang, who is very good indeed at math (top 15 in the Putnam his best year) took it, disliked it because it was boring recipes, and forgot all of it in a year. I tried it at community college a couple of times, and quit both times because it was boring recipes.
For those who loved diff eq, and for those like me who love math but find math recipes boring, I highly recommend [The Teaching Company’s Differential Equations, the Visual Method](http://www.thegreatcourses.com/courses/mastering-differential-equations-the-visual-method.html), which has a different approach. Nowadays we have computers to find analytic solutions for the very few differential equations for which analytic solutions are know, and numeric solutions for the rest of them, so recipes alone are of little value. The visual method instead gives mathematical intuitions about how differential equations behave.
Don’t be fooled into thinking this is an easy course, where you can just sit back and watch. Mathematics is not a spectator sport. To get through this class, you’re going to have to work through the mathematics. Mr. Fang can just watch the lectures and understand the material in real time (mostly) but mere mortals are going to have to treat this like a real course and work through the math and the examples on paper, and use the pause and replay buttons a lot. But those who do are going to get a good qualitative intuition about differential equations, such as is not gained from a typical differential equation recipe course.
"I’m not 100% sure that the next course needs to be or should be in the Calc sequence (Multivariable or Diff Eq)… I think there is some breadth to be gotten as well, like maybe Topology, other Analysis courses, Probability/Statistics beyond the AP Stat curriculum, Data Structures, etc.
…exploration and breadth are also really important…As a somewhat mathy person myself, I love the idea that a smart high-schooler can be thinking about other kinds of math in addition to learning more techniques to apply if/when s/he is an engineering major."
Gee, fret, where’ve you been all my life?
Have to agree with this 100%.
Just so you know, some of the top universities will tell you that even if you have taken upper level post BC classes like multi-variable before you get to the university, you will take it again there, period. Why? Because if you want to walk around with a mathematics or physics degree that bears the name of that university, you will learn math “their way.”
One way to get around this if one cares about taking the next course in the sequence or few without the credit is to make an appointment with the departmental chair and/or Prof of the more advanced class. Once there, do your best to convince them to permit the student concerned to skip the repeat and proceed directly to the next advanced course or few in the sequence.
Had no trouble doing this for many of my undergrad courses.
It may differ whether it is taken at a high school without college credit, or at a college for transferable college credit. It is not a given that subject credit will be rejected; one poster mentioned in the past that her student received subject credit from MIT for calculus and higher math courses taken at a California community college and/or local state university while in high school, but that MIT did do individual evaluation involving review of textbooks and syllabi.
Also, any college that admits transfer students will have to handle transfer students’ course work and determine equivalency. It does no good to admit a transfer student and then tell him/her that s/he has to retake several semesters of already completed work.
Some colleges’ math departments have their own subject credit by departmental exam procedures as well, for students who know the material contained in a math course but do not have any college (or AP/IB) credit for it. An analogous situation would be found in foreign language departments, since many students enter with some knowledge of the language (either as heritage speakers, or from high school course non-AP course work) and need to be placed in a higher than beginner level course.
Ahhh, but this is the perfect time for the student to go ahead and get that exploration and breadth that fretful spoke of. That is why I mentioned it.
Good to know, though, so thanks.
We learned otoh that many colleges that require most every student to re-take classes such as multivariable calc or diff eq unless the student is insistent upon proving they are ready for the next class, can be very strict about forcing students into a higher level of foreign language than they feel ready or willing to take, based on the results of a placement test or on “heritage speaker” status.