So much varies based on the child and the school. Oldest is now reading Math & CS at Oxford. He was only accelerated one year by his school (Geometry - 8th; Alg II/Trig - 9th; pre-Calc - 10th; Calc BC 11th; Stanford MVC - 12th) and got a one-on-one tutorial in 10th grade geared towards AMC/AIME. He took other opportunities like HS math team and an outside math competitions (HMMT/PuMAC) to feel challenged. That approach worked for him. There was no need for us to push him or the school to accelerate more.
To that point:
For the typical school, that is 2 years accelerated
This is actually the +2 math track. The regular +0 math track has algebra 1 in 9th grade, geometry in 10th grade, algebra 2 in 11th grade, and precalculus in 12th grade.
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The colleges you listed and most others frequently discussed on this forum require that students take a math placement test to help determine appropriate math starting point and sequence. You can get some idea how many are skipping math classes based on enrollment figures at the different starting points.
For example, Stanford students take a math placement diagnostic which recommends starting in Math 19, Math 20, Math 21, or Math 51. Everyone who wants to enroll in these courses at Stanford needs to take the placement test, regardless of the AP exam score or math background. The overwhelming majority of Stanford students seem to skip over math 19-20, which covers basic derivatives and integrals. The majority also seem to skip over math 21, which concludes single variable calculus. Instead most students seem to start at Math 51, which is the highest math level that the placement test may recommend.
Math 19: ~250 (Single variable calc: Part 1 â Limits and Derivatives)
Math 20: ~350 (Single variable calc: Part 2 â Integrals)
Math 21: ~900 (Single variable calc: Part 3 â Special series, Unbounded functions)
Math 51: ~1850 (Standard Multivariable calc: Part 1)
CME 100: 186 (Standard Multivariable calc using Matlab Part 1)
Math 61CM: 89 (Advanced Multivariable calc: Part 1)
Math 61DM: 31 (Advanced Multivariable calc: Part 1)
While not common, itâs possible to skip over multi-variable calc and start at higher levels. I did this, which in retrospect was probably a bad idea. I expect I would have had a more rigorous math foundation, if I repeated the multivariable calc at Stanford. Stanford also has options for special versions of the class, for students who have mastered basic multivariable calc, such as the math 61CM and math 61DM classes mentioned above, as well as one target for engineers that uses Matlab.
In contrast the single-variable calc math 19-21 are not rigorous classes. They might be thought of as remedial classes for students with relatively weaker HS backgrounds. Stanford used to offer a math 40-41 sequence, which covered the classes at a more standard pace (2 quarters in stead of 1 year), but discontinued it recently and now only offers the slow version of single-variable calc.
Yes, agreed. I think many on CC believe that if a class was taken in high school there is a check mark and it will not be taken again. Very unlikely any kid is going to sail through a highly advanced college math course. Taking classes with some repeat elements is fine. Better to have a solid foundation so that when classes get really tough you have the basics.
Stanfordâs AP credit listings suggest that they consider calculus AB = 19+20 and calculus BC = 19+20+21 in terms of material content, although Stanfordâs math placement tests are required regardless of AP scores. So 19, 20, and 21 are not really âremedialâ, but Stanford is selective enough that most students are more advanced than those courses already. The faster versions 40 and 41 may not have much enrollment because the stronger in math students are already ahead of that material, while those few for whom calculus is new to them may do better at standard pacing (19, 20, 21) instead of faster pacing (40, 41).
Unlike many other colleges, Stanford does not appear to have math courses lower than calculus like Princetonâs MAT 100, or âcalculus with review of precalculusâ like Harvard Math Ma and Mb.
Actually, it seems more common for posters here to recommend unconditionally repeating calculus even after an easy A and 5 on BC when one starts college as a frosh. But the opposite seems to be the case when a student finishes calculus BC in 11th grade or earlier and wants to take additional math at a college while still in high school. In that case, no one seems to recommend repeating calculus at the college while still in high school.
In any case, there are some advantages to getting on the +1 math track, since a student who starts in calculus 2 (or 3) instead of calculus 1 in college may be able to shorten a long prerequisite sequence for some majors. But there are diminishing returns for math acceleration at +2 or more.
Maybe âremedialâ was a poor choice of words. I expect the overwhelming majority of Stanford students take calculus in HS (~95% of students report taking calc in Harvard freshman survey, and Stanford is likely to be higher with larger portion tech majors). Most Stanford students sufficiently learn calculus in this HS class or elsewhere, but a minority do not, as suggested by the relatively lower score on the math placement diagnostic. Stanford offers this slow-pace (compared to other Stanford classes) single-variable calc class for the minority of students who did not sufficiently learn calculus in their HS class. It may be âremedialâ compared to typical Stanford classes even though students taking the class are no doubt well above average compared to overall HS population. At less selective colleges, a course like this might be the standard first math course that most students take, rather than a slower and lower level one like at Stanford.
One math professor at Stanford said they removed the faster pace 41/42 because a good portion of students in 41 were choosing to drop down to the slower pace 19/20/21 within the first few weeks of the quarter. The first course in both single-variable calc sequences had relatively little enrollment. In the final year of 41, the totals were 156 students enrolled in 19 (slower pace), and 129 students enrolled in 41 (faster pace), compared to ~2000 in math 51 variants.
Stanford has a summer online bridge program for students who fear that they are not prepared for single-variable calc in Math 19. Some quotes from the website at https://mathematics.stanford.edu/soar-mathematics-program are below.
Stanford also offers other summer bridge program that target groups that may benefit from additional support for a smoother transition to being a freshman at Stanford, as described at Log in .Math seems to be emphasized more than other fields at many colleges, including Stanford. I expect this relates to math being a foundation subject for which it is difficult to catch up without external support. Many students who start out behind due to a weaker HS background choose to drop out of math-heavy majors like engineering, rather than try to build their math fundamentals and catch up.
I believe that MIT also only allows placement. I knew a young man a few years ago who got into MIT, had many AP classes with scores of 5, AND a college transcript from the flagship state U with A grades in these dual enrollment classes.
MIT said, âYou can take the final exams for our classes now and place out of those classes if you do well, but we wonât give you college credit.â He felt it was absurd to take final exams, since some of them would have been for AP classes that he had taken in 9th, 10th, and 11th grade.
MITâs AP credit policy is here: https://firstyear.mit.edu/academics-exploration/ap-and-transfer-credit/advanced-placement . Summary:
MIT used to allow a 5 on BC to get credit for and place out of 18.01, but now allows credit for and placing out of 18.01 with a 5 on BC and a passing score on an MIT math placement exam.
MIT allows a 5 on both physics C to get credit for and place out of 8.01.
MIT allows a 5 on either English to allow a broader choice of communication-intensive HASS courses for that requirement.
MIT allows a 5 on any humanities or social studies AP to get some free elective credit.
MIT gives nothing for AP scores in biology, chemistry, environmental science, computer science, or statistics.
Incorrect. MIT has internal exams for MVC, LA, and DE. Passing them earns credit.
But they have internal exams for CS, chem, and bio, as well as physics for 8.02, 8.03, and 8.04, all of which give credit.
Iâm not entirely certain what counts as +0, +1, et cetera, but going by @ucbalumnusâs definition upthread that the +0 track is precalc 12th grade, that means my D19 was +1 at the beginning of 11th grade (took precalc that fall), and +2 at the end of it (took AP calc AB), since her school was on a block schedule and she doubled up on math that year because she consistently did well on it, despite always feeling very unsure of herself and like she wasnât quite getting it (which, along with her being an inconsistent standardized test-taker, is part of why she opted to not take the AP calc exam).
12th grade she did dual enrollment for most of her classes, and she placed into precalc using the local collegeâs placement test, ending up 1 point shy of placement into calc I every time she took it. (They use the ALEKS test. If sheâd been there one year earlier sheâdâve placed into calc I based on her ACT math score, but for whatever reason they stopped doing math placement based on SAT/ACT scores.) She was, to say the least, disappointed.
However, she ended up taking DE precalc fall semester 12th grade, and then calc I spring semesterâand now, as a successful industrial engineering (read: all math all the time) major and math minor, she sees that as the best thing that could have happened to her. Retaking precalc got her to actually learn why trigonometry works the way it does, for example, rather than just knowing what all the formulas are and how to use them. Sheâs convinced, and I suspect sheâs right, that sheâd still be doing as well in all her math classes now if sheâd placed directly into the DE calculus course fall semester 12th grade, but she wouldnât understand it as deeply, and she wouldnât love the subject the way she does. (And she does. I had only ever heard one person wax quite so rhapsodic about the joys of partial differential equations before her, but sheâs in love with it.)
I donât know what this says to the wider conversation, and this is just one example and generalizations from anecdote are always risky at best, but the TL;DR is that if her experience actually can be generalized, a thorough grounding in the (advanced) fundamentals is good, and repetition isnât bad.*
* Probably better: Repetition isnât necessarily bad. My own personal experience is substantially different from hers but still requires such a caveat, but this post has gotten long enough as it is, so Iâll forbear.
Seems like the high schoolâs math courses were of inferior quality.
Repetition may be necessary for some students because of inferior prior teaching (e.g. high school math through precalculus that does not adequately prepare students for calculus), but it would be better if there were not inferior teaching that requires some students to have to repeat what they were supposed to have learned before.
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Actually, no, given the track record of students who took themâthat would be an unwarranted conclusion.
What kind of track record? Do most students who complete precalculus at that high school need to repeat precalculus later, or not?
Generally not, and most students from the school who take the calc AB testâwhich is a small number, but itâs a very small high schoolâget a 3+ on it. My daughter simply, as mentioned in my previous post, doesnât do well on high-stakes tests (which was her perception, rightly or wrongly, of the placement test).
Our familyâs experience was that Algebra 1 was the critical class. If it was a full, very rigorous course, taught in depth, that laid a foundation for everything else.
S1 was extremely accelerated in math (majored in it in college, too). Had extraordinary teachers in pre-Alg and Alg 1 at STEM selective admit programs. Calc BC was one semester and they kept going. Stat was calculus-based. Got through much of an undergrad math curriculum in HS (MVC, DE, LinAlg, Discrete, Complex Analysis w/proofs). Back then (2005-2008), very few schools offered these classes and the colleges knew how kids from this particular program performed, so they tended to be generous with placement. S1 was never worried about the difficulty of math or CS classes. Didnât need tutors, didnât spend hours on homework. He is very, very good at teaching himself, but every acceleration was at the initiative of the school.
S2 was at a similar program for the humanities. His Honors Alg 1 teacher didnât cover nearly as much material. S hit the wall in Alg 2, when I had to buy an old-school Alg 1 book to teach him what he didnât get the first time. This was a kid in IB with outstanding math scores.
The teachers at both my kidsâ programs complained that in the rush to get lots of kids through Alg 1 in middle school, the curriculum was watered down to the point that the Alg 2 teachers had to do lots of remediation. The difference between S1âs AP Calc and Stats classes and S2âs was night and day. They were two grades apart in school and the revised Alg curriculum was installed after S1 took it and the year before S2 took it.
Itâs all about the foundations.
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I found Art of Problem Solving âAoPSâ when D was in 5th grade. She worked through the preAlg, intro to Alg, Intermediate Alg, precal and calculus. This has given her a good foundation in math . Of all these texts, I really feel the PreAlgebra was the most important. It made Algebra pretty much a cake walk.
Hello from the other side!
I have a gifted ELA kid, dd22. She thinks because she is on the regular track in math that she is in âdumb people mathâ. She is taking Math III in 11th grade.
There is a lot of pressure (from peers, parents, society) on the kids to accelerate in math so that they can get into a âgoodâ college. My kids are just not math kids. They are bright kids, but their brains are more geared toward language (English and foreign languages, history, geography, arts). Their peers, however, jumped into accelerated maths and many would take courses simultaneously (take Math II in class in 9th grade, and Math III virtually online at the same time so they could be in PreCalc in 10th grade). I donât know if their parents pushed them into this or they pushed themselves. But it is a LOT of kids at my daughterâs school. I would say at least half her class is accelerated by at least one year. Itâs a small school, but that is pretty much the expectation if you are a âsmart kidâ.
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