Let me say one last thing on this topic before moving on. There is often a visceral dislike / fear about Math that I don’t understand. In fact, APUSH, is a far harder course than any high school level math course. The amount of material you need to remember is truly stupendous. Yet no one debates whether they should take APUSH or not. They either take or they don’t. There is no drama/excitement in the decision. Whereas Calculus, which is really just slopes, areas and limits, takes on a life of its own. Sometimes parents tell kids – oh, let’s think hard about this. Do we need to really do this? It’s not a big deal. It is just slopes, areas and limits :-). It is very intuitive. It is very visual. Much easier than APUSH. There aren’t that many things to remember. Math (even Calculus) is in fact a lazy person’s choice compared to much harder subjects like history or biology where you need to memorize fat books.
In our household, calculus was a given and both kids thought about whether or not to take APUSH. This is very common in our area since APUSH is a tough class so I am not sure why your experience is so extreme.
Your post seems to imply a universal aptitude alignment which doesn’t exist in the world. Some people are naturally more oriented toward math or science concepts and others toward other learning like historical facts. Just because there’s more to memorize doesn’t make it harder if someone has an easier time with contextual memorization but a harder time with logic or math concepts. Saying APUSH is harder as if that’s an objective fact while Calc is a “lazy person’s choice” really just says that you’re probably a math-oriented person and that’s your mental frame of reference. I can assure you my daughter found APUSH “easy” and got a 5 but didn’t find AB Calc easy at all and got a 3. Just different minds.
It depends on the person. Many persons find subjects emphasizing numerical calculations to be far more challenging than subjects emphasizing memorizing and regurgitating facts. I think people are more likely to fear that they won’t be able to handle numerical calculations than fear than they won’t be able to handle memorizing facts.
Heh. At my son’s high school STEM program, former students brought back problem sets and tests (honors calc, MVC) from their top-flight colleges and the teacher used them as problem sets for the HS kids. Analysis 1 covered BC calc in a semester, and they kept going deeper. AP Stat was taught with Analysis 1 as a pre-req – all calc-based. MVC, LinAlg, DiffEq and Complex Analysis (with proofs) were all one-semester courses.
He ran the course catalog on CS classes, was a TA and coached folks competing in USACO. Most of his CS was self-taught, as he had been interested since an early age.
Mathematical Physics (taken in lieu of AP Physics Mech and E&M) used UMD’s honors physics major syllabus. MVC was pre- or co-requisite. The AP exams and USAPhO exams were pretty easy after that class.
He placed into Honors Analysis at a T5 and got credit/placement for two years of math, and took graduate theoretical CS classes freshman year. Majored in math, mainly to improve his CS toolbelt. Decided against a PhD in theoretical CS, as he found his internships offered opportunities for theory and other things up his alley.
We were extremely fortunate that our public school system offered these opportunities, and that some extraordinary teachers helped nurture his talents and accelerated him as appropriate. The beauty was that there was enough of a critical mass that he could take these classes with kids his own age – and he wasn’t the top math student there.
A student shouldn’t treat calculus, or math generally. as numerical calculation. Nor should s/he treat APUSH, or humanity subjects generally, as memorization or regurgitation. If s/he does, s/he isn’t doing it right.
Math acceleration in terms of college admissions is viewed through the lens of what is available at your HS. Despite the fact that most MA public HS’s don’t offer courses beyond AP Calc AB/BC and AP Stats, kids are routinely accepted into top schools. We send about 20-30 kids to T20s every year - including 1-2 to MIT most years --so the lack of additional acceleration doesn’t seem to hurt. As it is, most kids in the US don’t even take calculus in HS -the +2, +3 tracks discussed here simply aren’t available most places. Obviously, for a lot of your kids (who seem to be real math whizzes) this additional acceleration was what they needed and wanted - that isn’t true for all kids. A close friend has their Phd in math from Stanford and isn’t a huge fan of acceleration on a large scale - they believe every topic can be explored more deeply and that some (not all!) kids are treating math like a race to be finished as opposed to a topic to be deeply understood and explored.
All students mainly pre-med and stem repeated the math classes. He said some of the students “audited” and took the tests to be familiar before they took the class for credit. At his school the math department is notorious for being extremely difficult.
The school does allow AP credit to be used for math if you receive a 5 on the exam. There really is no warning for students before they start taking the calculus, diff. eq., or linear algebra how the instructor will structure the class. Additionally, each instructor “creates” one test for the whole department each semester. So, for instance, he had a Russian prof. who was more interested in one particular theory/concept and spent a majority of time on it. However, the exam was made by another professor so… his students were at a disadvantage. You would think that after student reviews they would change how they do things but this has been the case for a long time.
Our third son was very diffident in Algebra II. He is in advanced pre-calculus, but did horribly on the ACT practice exam. I was honestly shocked at how low his score was on the test. He has been working with a tutor and “relearning” ALG II this semester through Saxon Math. Some of the concepts he never had in geometry or ALG II, probably because the covid curriculum was not as intense.
I really agree with you. Our older daughter, who enjoys the learning process, mentioned several times in high school she wished the classes would slow down so she could “really go deep in particular topics.” She always felt like they were skimming the top. The summer after her first year in college, she went to a top university to take a summer more theoretical math class. It was hard but she enjoyed the rigor.
I support dedicating resources to math (and STEM, generally). Advanced students should also be provided with learning and development options whenever possible. Where math acceleration can become problematic is when we sacrifice in other areas. Exposure to music, art, theater, foreign languages, history, civics, and other social studies is woefully inadequate in many American schools and has been made even worse by overemphasis on STEM training, such as movements to double-up on math classes late in grade school / in middle school.
My daughter attends a good, not amazing, public high school in PA and it’s standard for the high honors kids to take Calc AB in 11th & Calc BC in 12th. I can’t remember what year it was (maybe 5th or 6th?) that they had two math periods. In retrospect, I wish she would have had more art and music and the option to start a foreign language a year earlier, rather than more math.
I agree, it makes more sense to me when acceleration is done early enough so there is never a need for 2 math courses in a year. Schools here do it between 4th and 5th and the +2 kids just take a different math than the rest in 5th(they call it “advanced”). Between 5&6 or 6&7, the “regular” kids split again to become the +0 and +1 kids, or some +2s can slide back to +1, and occasionally some +1s move to +2–but they just do some extra work with a teacher in the summer, for a few sessions. (The rare +3 just take 5th grade math with the 6th advanced kids–no double math either, just a few extra sessions with the teacher). The only kids that take two core math courses in a year in order to accelerate are the kids who transfer in later from a school that didn’t have +2 acceleration, and they take geo and alg2 together. Ideally, math acceleration should not take the place of other subjects!
It can work well. One of my kids skipped a grade. Teacher called the second month of class. My kid was throwing off the other kids. Worked out great for us.
This is very true. There are not many schools which offer +2/3 advanced classes in high school. Two things we found, if a private school has international students it can be offered (as many international students are by design a few years ahead of the US). Another case is if a school is a magnet pubic or very large public with a feeder from an advanced/gifted program +2 can be offered. Most high schools end in Calc. Some don’t even offer Calc or send students to online programs/CC/local colleges for it.
I’m not of the opinion that taking advanced classes beyond +1 is a recipe to elite colleges. But I am a firm believer that boredom is insufferable. So, if a kid is motivated has great math skills and has an available path then I’d let them proceed. I think there’s a slight advantage when they get to college in that, they’ve already seen some of the material.
It should be noted however for the subset of students wishing to pursue STEM studies outside of the US, many countries do require incoming students to have taken Calculus in high school.
This is often the sign of someone who is gifted. The kid in K, who doesn’t just want to know why the tadpole grows legs, but how long does it take, and how does it work and so on. They want the entire picture. Some even want to connect it across subjects.
Your daughter will likely do really well in college. When looking at colleges we try to see if there is cross subject collaboration in areas of interest. Some colleges run like silos. You cannot take a business course unless you are in the school of business or you cannot take it until it’s filled. That can be frustrating for a kid who delves deep and follows their passions. My oldest wrote a paper which led from one thing to another. It opened up three more areas of interest.
Unfortunately, the educational system isn’t set up to support this type of rigor. Instead we push kids until grade 12 to get a high GPA ( based on CC often at the expense of true challenges and learning). And we push kids to think that GPA is more important than the actual learning. And bringing in a dose of realism, kids have to pay attention to GPA. Oh well, I get so depressed when I think of the GPA races and math education in the US. But it is, what it is.
At our school, both my kiddos were accelerated in elementary school itself - 1 year ahead of the regular class. This continued till high school. They were also identified as Gifted & Talented - which meant nothing in practice other then getting a first shot at some competitions. They allowed them to skip Calc AB to take Calc BC directly.
My son, this senior year, took College in High School class in Linear Algebra and AP Stats.
She did do fantastic in college even took a couple of grad level advanced engineering courses. She is working at an aerospace company and thinking about pursuing a masters or perhaps going into research.
Yes, those type of students can really fly 
once they get into a program that supports their needs. Both aerospace and research sound great.
If kids have a great deal of enthusiasm for Calculus, I am told books of the kind I list below offer a better, more robust foundation. You can do them either after you finish calculus in school, or parallelly so that you understand the material better. I am sure there are other books like this – these are just two examples:
https://web.math.princeton.edu/~gunning/bk.pdf – here is a review of this book: An Introduction to Analysis | Mathematical Association of America
I wouldn’t call them an easy read though …
It’s a crazy thought, but perhaps most kids (especially math kids) might be better off studying “Accelerated Honors Analysis” with Dr. Guning when they get to Princeton, rather than before.