Calculus BC 8th Grade/Honors Science Middle School

Parents Forum
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@SatchelSF wrote

@vox_nihili I am a PhD mathematician and I would definitely suggest this as a step after Calculus BC. It’s much better than to track through the usual sequence of Calc3, diff eq, etc. - these tend to be more computational-leaning as taught in most high schools and colleges.
An advanced student should be able to pick up the material on their own at that point . Most of us self taught the math at upper levels, because the profs often mumbled and there was really no other way to learn the learn the material even if some profs delivered excellent lectures . Much different than the “show me” math of earlier courses.
Prof. Francis Su of Harvey Mudd has a nice series of lectures on real analysis - the video is not great quality, but his explanations are very clear.

I would encourage your daughter to look at contents like AMC - at least to try the problems. AMC8 is in November. My DS2 takes the AMC contests to see how he fares relative to the national group. He doesn’t feel any pressure to be uber-competitive. And the experience of solving the out of the box competition problems is very valuable.

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Mom of a math major here.

S1 had radical acceleration in math and fortunately was at a STEM HS program that offered post-AP classes. They joy was that he was able to take academically appropriate courses with his age peers. Having that cohort of kids all plugging away at a shared passion was critically important.

He took Alg I on steroids in 6th (as in they also covered half of the usual Alg II course), Magnet Geometry (with proofs) in 7th, Functions (2nd sem Alg II/PreCalc/Trig and some AB Calc) in 8th – and that is where he hit a wall. The Functions teacher wisely noted that he was still 12 and one’s ability to handle abstractions develops a little later. She required a B+ to stay in the class (he had a B), and so she booted him out of the class. No math the second half of 8th grade. He had her again for Functions in 9th and aced the class. She was right – he needed that extra year to mature and develop the ability to consider the abstractions. Having a deep understanding of the foundations is critical. Acceleration isn’t enough.

He didn’t do CTY, AOPS or Stanford math. If we lived elsewhere, we would have encouraged it, but there was enough offered at HS or that he did on his own that kept him engaged.

At the STEM program, he went straight to Analysis I, which was BC Calc with lots of additional stuff (and if I understood the math, I’d tell you what else they covered), taught in one semester. He also took Multivariable Calc, Discrete, Real Analysis (with proofs), Lin Alg, DiffEq & calculus-based Stat. All of these were taught as one-semester courses using college texts & problem sets. Alumni would come back to visit and they’d bring copies of their problem sets. S would bring home packets still labeled as Math XXX from math departments at Michigan, Dartmouth, etc.

He did some of the math competitions, but in a desultory way. Didn’t care enough to fight for a top spot. He saved that intensity for computer science. He did a summer math program after soph year where he derived lots of proofs. Rocked his world and frustrated him simultaneously.

He also took many post-AP computer science classes in HS (algorithms, combinatorics, data structures, and a few more I don’t remember), Economics, and Mathematical Physics, which required Multivariable Calc as a pre-requisite and made the AP Physics C exams easy. He looked for opportunities in other departments to expand his math horizons.

One of S1’s programming teachers from middle school suggested that he get involved in USACO (US Computer Science Olympiad). It has excellent free curricula and is algorithmically-based programming (the proper way to do it!). There are several competitions during the year (at various levels depending on skill), lots of encouragement and the opportunity to compete for a spot on the US team that goes to the international competition. This was the best thing that happened to S. He self-taught programming, but USACO taught him to be more efficient and to solve the problems mathematically. He also learned a lot of theoretical CS, which is his real love.

He wound up majoring in math, mainly to support his computer science. Having both sets of skills in his tool belt has made him eminently employable. :slight_smile: Always a good thing!

OP, the big question is how supportive can the school realistically be in getting your D the level of work she’s capable of handling. What if she decides to go off the intense math path? What classes will they offer if the number of kids in her accelerated cohort decline to pursue other interests? Is taking courses at a nearby college an option? There were some students at S1’s program who went to our flagship right down the road for math courses, but he felt more comfortable socially in the magnet program. Junior and senior year he studied with a prof in CS/math at the flagship after school as part of his senior research project. That worked out really well for him.

P.S. S1 talked to Prof. Su a couple times when he was deciding on colleges. Su is great, and very accessible.

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OP, I think the least of your concerns should be College credit. If she is taking BC calc this young, and doing well, it seems dual enrollment or independent study is in her future. I suggest finding a gifted resources person, privately if necessary, and talking to that person.

Also, a cautionary tale… one of my son’s friends is a kid like yours. Got pushed a bit too far a bit too fast and ended up dropping a level in math one year to fill in the deficits. Granted this problem was partially due to our school’s ineptness on seriously advanced middle scoolers, but still. Your kid may be bright, but she is still young.

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As to public versus private school, that will depend a lot on the particular public and private schools available to you. My mathematician daughter has been in a small private school, homeschool, a public STEM magnet and full-time DE at Georgia Tech. The public school is what allowed her to do dual enrollment (after some wailing and gnashing of teeth/fighting with the GC), and it was almost free in Georgia. When she was in elementary school, a gifted counselor at JHU’s program told us that lower-ranking public schools often have a “we try harder” mentality and are actually more willing than elite private schools to accommodate highly gifted students. Also, the public school doesn’t have the option private schools do of kicking you out if you ask too many questions. I’ve been threatened with my kid’s being kicked out of a private school on more than one occasion and by more than one school (good times!), so this one was important to me. Whatever you do, do not rely on any GC over doing your own research. In our experience, GCs are there for the kids whose parents have no idea how college works, not for parents who are capable of and do their own research. I have been less than impressed by any of the four GCs, at three different schools, that my kids have been assigned. You’re your kid’s best GC.

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@vox_nihili - a few other random thoughts from a parent of a mathy kid.

If she hasn’t already, as a rising 7th grader your daughter should consider taking the SAT this fall for a number of reasons. It should be pretty easy for her to qualify for SET through Johns Hopkins CTY (you just need over 700 on one of the sections, and the math is trivial) and then you’ll get two things: (1) a SET counselor who can offer some advice (not transformative in our experience, but still helpful) and (2) you’ll get all the invites to local SET gatherings where you can meet other kids and their parents. If you are in California (sounds like you are), there are a lot of these kids; we found these gatherings tremendously helpful and our kid made some good friends through them.

You’ll also be able to qualify for Davidson Young Scholars and also Davidson Academy Online classes if that seems worthwhile.

An SAT score should also help if she wants to take any Stanford OHS online courses. These are expensive, but very good from what we understand from other advanced kids (ours was accepted at SOHS but ultimately enrolled in a private high school after K-8 in public school).

You should also look at the Caroline Bradley Scholarship for high school, which is a fantastic program, and only open to 7th grade applicants. This program requires either an SAT or ACT score.

Regarding simultaneously doing AoPS courses (or at least working through the texts) while taking regular classes with her cohort at her regular school, AoPS Precalculus this year will tell you a lot. I bet it will be quite a lot more rigorous than the precalc she will be doing in school. If that all goes well, she might want to consider supplementing BC Calculus with a more theoretical treatment (BC calc is basically just a plug and chug course). We don’t have personal experience with AoPS calculus, so maybe that would be great, but I think I would lean towards supplementing with something like Spivak’s Calculus textbook. That is a terrific treatment - not too difficult for a beginning student - but miles ahead of any book that will be used in BC by a high school. Others also swear by the Apostol text for the same purposes, but we don’t have any personal experience with that.

Potentially look into math circles in your area, which are pretty big in middle school. Although this might not be applicable where you are, our high school kid established some contacts with a local university math professor through math circle, which can help smooth over dual enrollment issues later on.

Last, you will often hear that girls are not welcome in math, but we personally haven’t seen that from other math students and teachers. What we have seen is social pressure - sometimes pretty intense (or at least perceived that way at this age) - by other girls to not “hang out with the math nerds in Mu Alpha Theta…” so that is something to watch out for as time goes by.

Again, best of luck!

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This may point to a benefit of an all-girls high school. My rising senior D is an officer in her Mu Alpha Theta club at an all-girls school. They don’t win tons of competitions, but the girls are very supportive of each other’s successes. It’s also a nice social opportunity as nearly all the top students happen to be in this club. It’s also a good chance for the girls to know the top students in the grades ahead of them and hear about their college app paths.

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So much great information. I would second what a few posters have mentioned, know all of the options and weigh what is best for your child. Raising gifted kids is difficult ( and expensive!) . Many a multifaceted gifted and want to be great in all areas. There is just not enough time in the day. We have done multiple programs based on their interests and needs. Kiddos often tell me when they are getting overloaded ( for example a specific person who tutored high level kids for Nat’l math contests). Kiddo #2 just wasn’t interested in doing this, found it boring and chose to do another program.

In terms of the girls in STEM/Math. There are often discussions about this. I see it as a world of opportunity. There are few women in many fields and the sky’s the limit. Some look at it as difficult, I think the glass is half full. In just five years, I have seen the number of girls at National and World level math/STEM related events changing. While it’s not always 50/50 it is changing and growing. Not interested in debating back and forth about all the negative aspects, what I am saying is to be open to the opportunities for women. There are many.
Wow, I am jealous there are schools which teach middle schoolers real math. The only options we have found were online or outside of school. My kids sat in math classes for years bored to tears. What a waste.

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In addition to Spivaks calculus book, I would also recommend Victor Bryant’s “Yet another introduction to analysis” and Kevin Houston’s "how to think like a mathematician ".Both are accessible, with excellent exposition and motivation for underlying ideas. Cambridge university press has other math titles like these.

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While we’re talking book recommendations, this may not be applicable for the OP, but for anyone else who’s lurking – Will put in a plug for Jacobs’ Elementary Algebra. It is SO much deeper and thorough than the usual Alg I textbook students use these days. We used it as a resource when S2 was taking Alg 2/Trig. S2’s math teacher, who went to MIT, recommended it. Good stuff to reinforce depth. Jacobs also has a Geometry book which also got great reviews from my math-intensive GT friends. It actually includes proofs, which have largely disappeared from the geometry curriculum (and are critical to math majors).

S1 also loved Spivak. UChicago used it for Honors and IBL Analysis when S1 was there (ten!!!) years ago.

Most importantly, let your D take the lead here. Clear the obstacles, but let her create the path.

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Thanks, @CountingDown Those are great books for algebra and geometry. For those who have kids who think they hate math, but love reading, I would add Harold Jacobs’ Mathematics: A Human Endeavor

On a separate note, it’s really cool to be discussing all this math stuff!

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For those who keep saying that math contests are good here is what the best mathematicians in the world say about them including at least three Fields medalists.

[In a mathematical examination of high standards and traditions] the examiner is not allowed to content himself with testing the competence and the knowledge of the candidate; his instructions are to provide a test of more than that, of initiative, imagination, and even of some sort of originality. And as there is only one test of originality in mathematics, namely the accomplishment of original work, and as it is useless to ask a youth of twenty-two to perform original research under examination conditions, the examination necessarily degenerates into a kind of game, and instruction for it into initiation into a series of stunts and tricks.

— Godfrey Harold Hardy

These [mathematics] contests are a bit like spelling bees. There is some connection between good spelling and good writing, but the winner of the state spelling bee does not necessarily have the talent to become a good writer, and some fine writers are not good spellers. If there was a popular confusion between good spelling and good writing, many potential writers would be unnecessarily discouraged.

— William Paul Thurston

Professional mathematics is not a sport. In particular, whether one is “better” than one’s peers is not really the right thing to focus on; the more important thing is to ensure that one can do good mathematics in one’s chosen research area.

— Terence Tao

What is important is to deeply understand things and their relations to each other. This is where intelligence lies. The fact of being quick or slow isn’t really relevant. Naturally, it’s helpful to be quick, like it is to have a good memory. But it’s neither necessary nor sufficient for intellectual success.

— Laurent Schwartz

There was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates, almost by sleight of hand, the most forbidding subjects.

— Alexander Grothendieck

The most profound contributions to mathematics are often made by tortoises rather than hares.

— William Timothy Gowers

I tend to be slower than most mathematicians to understand an argument.

— Stephen Smale

It’s true that I’m not good at solving problems. For example, I would never be good in the Math Olympiad. There, speed counts and I am certainly not a speedy worker. That’s one pleasant thing in mathematics: it doesn’t matter how long it takes if the end result is a good theorem. Speed is an advantage, but it is not essential.

— John Tate

Let me stress that creating new mathematics is a quite different occupation from solving problems in a contest. Why is this? Because you don’t know for sure what you are trying to prove or indeed whether it is true.

— Sir Andrew Wiles

Problem-solving should never be practiced for its own sake; and particularly tricky problems must be excluded altogether.

— André Weil

In one way I think my early involvement with Mathematical Olympiads and suchlike has damaged my mathematical taste: I still have this feeling that short, elegant problems should have short, elegant solutions, and that one should be able to do mathematics without absorbing a lot of difficult stuff first. Unfortunately, this is all false.

— Gareth McCaughan

I was never interested in artificial puzzles.

— Goro Shimura

I was never fascinated by puzzles or intellectual games.

— Robert Phelan Langlands

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@collegedad13 I think done well math competitions can spark interest in some MS/HS students. It’s just another way to
explore solution to problems. Yes it is possible to get caught up in the competition part to an unhealthy degree.

Some like USAMTS are done individually and not timed. No it is not the same as math research, but aside from number theory it is the very rare student who is capable of math research before undergrad (even top students who take college classes in HS).

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And yet, @collegedad13, didn’t about half of the recent Fields medalists (say, over the past two decades or so) compete in math contests? Terence Tao himself was perhaps the youngest IMO gold winner ever. And the only female Fields medalist ever was also IMO for Iran (Mirzhakani, sadly deceased now).

Besides, not every math contest participant has to become a pure mathematician for the contests to have value. For many, and Tao himself acknowledged this, the contests are first and foremost a bonding and social experience where some kids for the first time find their “tribe.” That’s gotta be worth something.

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@ma2012 there is a study out of Stanford that indicates that math competitions cause girls to go backwards in their math studies Niederle is the author and she says as follows

“Many factors may explain why fewer women end up completing math-intensive
college course work. Partial explanations may be found in examining the explanations women give for dropping out of these courses. A report entitled “Women’s ions women give for dropping out of these courses. A report entitled “Women’s Experiences in College Engineering,” funded by the National Science Foundation
and the Sloan Foundation, writes that the exit of many young women is not driven by ability, but rather that this decision is influenced by women negatively interpreting their grades and having low self-confidence (Goodman Research Group, 2002). Furthermore these women mention that negative aspects of their schools climate, such as competition, lack of support, and discouraging faculty and peers, cause them to reevaluate their field of study. In an earlier study of engineering student performance and retention, Felder, Felder, Mauney, Hamrin, and Dietz
(1995) find similar effects.”

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My son enjoyed Jacob’s* Math: a Human Endeavor* in late elementary school. He also really liked *The Book of Numbers * by John Conway and Richard Guy. https://www.amazon.com/Book-Numbers-John-H-Conway/dp/038797993X

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As another poster pointed out, this thread has been filled with great information! A lot of great suggestions and opportunities mentioned - courses, summer camps, contests, gifted resources, talent programs and more! I have a lot to research including looking into the policies of the local public high school. I will also have to research and discuss with the school administration a host of topics such as policies and procedures regarding dual enrollment, college credit and transfer credits from online programs such as Stanford Online and AoPS. Does anyone have good tips on how to advocate for one’s child? I am not a pushy person so I have always found it a bit uncomfortable to do so, especially when faced with pushback. However, a parent sometimes has to speak up when, as an example, an administrator will suggest a child would probably fit in better taking a math class where they can act as a tutor to their peers than being in a math class where the material taught will be new for the child. If I have to, I will advocate but I would like to know if there is a way to avoid such a suggestion from surfacing.

I’m also really grateful to all of you who shared your own personal stories of what it took for you to help your child navigate a path to succeed in their academic journeys. I also appreciated hearing from the people who successfully navigated their journey and wish them even greater success in their careers!

I completely agree with posters who think it’s important that my daughter be the one to forge her own path. I also appreciate the words of caution. Right now, we’ll see how Precalc goes. I expect she’ll enjoy the AoPS course given her past with the series but one never can be completely sure. If she enjoys the course and wants to take Calculus with her cohort, I won’t stand in her way. However, at the same time if she is at all hesitant, she could always take other courses such as Intermediate NT and P&C instead and wait for a year. She’s expressed interest in physics, computer science and chemistry but the courses she will take the year after will be for her to decide as they were this year.

Thank you for the Calculus book recommendations. I will definitely check them out. Does anyone have similar suggestions for physics, computer science and chemistry books? I am hoping for books that are accessible for someone with little knowledge but yet contains similar depth to the aforementioned Calculus books. If she is interested, she can peruse these books and hopefully they will give her an idea of the type of material she would need to be ready to tackle if she were to take a high school course in one of these subjects. Any other resources in these subjects would be much appreciated as well.

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another poster pointed out, this thread has been filled with great information! A lot of great suggestions and opportunities mentioned - courses, summer camps, contests, gifted resources, talent programs and more! I have a lot to research including looking into the policies of the local public high school. I will also have to research and discuss with the school administration a host of topics such as policies and procedures regarding dual enrollment, college credit and transfer credits from online programs such as Stanford Online and AoPS. Does anyone have good tips on how to advocate for one’s child? I am not a pushy person so I have always found it a bit uncomfortable to do so, especially when faced with pushback. However, a parent sometimes has to speak up when, as an example, an administrator will suggest a child would probably fit in better taking a math class where they can act as a tutor to their peers than being in a math class where the material taught will be new for the child. If I have to, I will advocate but I would like to know if there is a way to avoid such a suggestion from surfacing.

I’m also really grateful to all of you who shared your own personal stories of what it took for you to help your child navigate a path to succeed in their academic journeys. I also appreciated hearing from the people who successfully navigated their journey and wish them even greater success in their careers!

I completely agree with posters who think it’s important that my daughter be the one to forge her own path. I also appreciate the words of caution. Right now, we’ll see how Precalc goes. I expect she’ll enjoy the AoPS course given her past with the series but one never can be completely sure. If she enjoys the course and wants to take Calculus with her cohort, I won’t stand in her way. However, at the same time if she is at all hesitant, she could always take other courses such as Intermediate NT and P&C instead and wait for a year. She’s expressed interest in physics, computer science and chemistry but the courses she will take the year after will be for her to decide as they were this year.

Thank you for the Calculus book recommendations. I will definitely check them out. Does anyone have similar suggestions for physics, computer science and chemistry books? I am hoping for books that are accessible for someone with little knowledge but yet contains similar depth to the aforementioned Calculus books. If she is interested, she can peruse these books and hopefully they will give her an idea of the type of material she would need to be ready to tackle if she were to take a high school course in one of these subjects. Any other resources in these subjects would be much appreciated as well.

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There are free physics textbooks available. There are books for high school physics, non-calculus college physics for biology majors, calculus-based physics for physics and engineering majors, and more advanced physics courses. Lists are found at:
http://www.cabrillo.edu/~jmccullough/Links/textbooks.html
http://www.openculture.com/free-physics-textbooks
http://physicsdatabase.com/free-physics-books/

http://physics.berkeley.edu/sites/default/files/_/f17_textbooks_web.pdf lists UCB physics textbooks.

The regular calculus-based physics courses for physics and engineering majors (7A, 7B, 7C) use:
Giancoli, Physics for Scientists and Engineers
Tipler, Modern Physics

The honors courses (5A, 5B, 5C) use the following:
Kleppner, Introduction to Mechanics
French, Vibrations and Waves
Purcell, Electricity and Magnetism
Bennett, Principles of Physical Optics
Wichman, Quantum Physics (in SI Units)
Reif, Statistical Physics (in SI Units)

For computer science, consider the following UCB CS courses whose materials are all online for self-study:
http://cs10.org : general beginner-level overview of CS
https://cs61a.org/ : first course for CS majors
https://sp18.datastructur.es/ : data structures course (second course)
http://www-inst.eecs.berkeley.edu/~cs61c/su18/ : machine structure course (third course)
http://www.eecs70.org/ : discrete math and probability theory

The textbooks and courses listed above cover material that is probably more in depth and/or advanced than what you will find in most high school physics and CS courses. AP physics C is a subset of Physics 7A and 7B (though the AP course uses less advanced math). AP CS principles has CS 10 as one of models it is based on. AP CS A is a subset of CS 61B.

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For physics, I think what @QuantMech said above about the subject requiring some physical intuition and practical experience makes a lot of sense. At this stage, plowing through a college text is probably not going to be very helpful.

What I would suggest is spending some time working through Khan’s videos, which are pretty good. There is another site, The Physics Classroom, that is also very helpful. Both of these are approximately at the level of AP Physics 1 and 2 (or whatever they call the non-calc physics sequence today…). Combined with whatever she is doing in school, she should start getting a good handle on the subject.

For a readable, but complete reference once she has some command of basic calculus, take a look at Young and Freedman’s “University Physics with Modern Physics” - I like that book and would be curious to know what real physicists would think (QuantMech?).

Looking at the contests might be fun after she gets some exposure to the subject. The f=ma test is the place where most of these kids start. AwesomeMath has some decent online courses on it, start here with the first one appropriate for algebra-based physics kids: https://www.awesomemath.org/academy/online-courses/#tab-id-3. Areteem Institute also offers their version of lectures/problem sets, but I don’t have any experience with it. If she really gets into physics, down the road, AoPS just starting offering Physics WOOT so that might be an option.

Good luck!

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OP, I don’t know if the purpose of moving fast in math is for college admissions or for pure learning. What were posted here were stories, not statistics.

If for learning, collegedad13’s advice may be more valuable as his daughter was so advanced and achieved what she wanted and more I guess when she turned down stanford. None of my kids were advanced in school math as they all took the BC in their junior years. But they did some math competitors and both had a 7 on their highest AIME scores. This was back to 7-10 years ago. Now, a 7 in AMIE may not help you that much. For college admission, with their records, I don’t think that they would get in to the schools they got in back then.