^florida26, have you read the thread at all? This is a 13-year old kid who got a 94 in UC Berkeley’s precalc final exam without actually taking the course, and did many problems “in his head”, can intuitively derive, who’s self taught calculus “for fun” and loves rings… we’re talking a math prodigy here, not a B student.
No we are not talking a math prodigy. Self taught calculus to what extent??? I can teach any 13 year the definition of a ring. It is doing something with them that is the challenge. From the original post “If he takes a placement test now, he will get a B/A- in PreCalc” So yes I did read the thread
He would get a B/A- because he didn’t know what “simple harmonic motion” is. That’s a vocabulary issue, not a math issue.
Doing pre calc in 8th grade doesn’t make you a math prodigy. Sorry
It is four grade levels ahead, so what is your definition of “prodigy” here?
@florida26 You sound threatened. Don’t worry, the 13 year old’s accomplishments have no bearing on yours.
I’m a bit confused about where the student is now, but from my understanding I’d recommend sequence #5. AP stats is not calculus based and if he’s interested in the topic he can learn it more rigorously in college. The only concern I have about the sequence is that it has him taking “MV calc/Linear algebra” before real analysis. Classes that combine MV calc and linear algebra are usually the sort of “applied” ones that engineers, chemistry students, etc. take. This does not typically prepare for real analysis or abstract algebra which are very proof heavy. Is there an “advanced” MV calc class he can take so he’s better prepared?
isn’t he five grades ahead?(precalculus is the senior class for reasonably good, college-bound seniors and this kid is in 8th grade.)
MODERATOR’S NOTE: 4 grades ahead vs. 5 grades ahead is a matter of semantics and serves only to derail the thread. The kid’s advanced in math. Please keep on-topic with advice, and not go off on tangents. Thank you.
I hesitate to bring this up, because the child is only in 8th grade and many things can change between now and 12th grade. However, there may be something to be said for “What’s the rush?” Taking MVC/Linear Algebra/Differential Equations/Real Analysis while in high school may look fine on paper, but what happens when he gets to college? Many private universities will want him to learn math “their” way, and not accept post-AP credit. What happens then? He retakes courses that he’s already taken?
I would caution any student/parent to beware of the calculus trap and read [url=<a href=“http://www.artofproblemsolving.com/Resources/articles.php?page=calculustrap%5Dthis.%5B/url”>http://www.artofproblemsolving.com/Resources/articles.php?page=calculustrap]this.[/url]
@skieurope said what I was trying to say in a much more eloquent fashion
What happens when he gets to college? Faculty never once consider having him retake any course, in fact actually encourage him to start the graduate level sequences as a freshman. His love for and ability in math grow exponentially with the exposure to even more advanced concepts. He attends, understands and is able to present at high level math conferences. He applies and is accepted to extremely selective REUs and internships due to his very advanced math knowledge.
Will let you know how it affects grad school apps.
Very advanced math students are a special species and are treated as such by the research universities that they attend. They typically enter whatever level they were at before and work at the graduate level.
I am a bit afraid to jump in the fray but I would tend to agree with florida26 that my son is not a prodigy. He is merely a kid who likes math and I would prefer that it stays that way as in our house the p-word is considered to be a very bad word which we never use. No he probably cannot understand graduate level abstract algebra as in the Serge Lang book however he can comfortably do undergraduate level problems in fields and rings as he is given such problems as homework by the college professor who teaches the math class that he and 3 other kids attend instead of the regular school math class. I know that they are UG level problems as I took those same courses when I was in college and did those same problems. But even that doesn’t make him a prodigy. I do not like the whole notion of prodigy anyway. It adds nothing but pressure. We believe that a kid should do math or anything else for fun and that’s what our son does and we like it that way.
At this point really what we have to decide is which course he takes in 9th grade which will be either Honors Precalc or AP Calc BC. Florida26, rest assured that we won’t let him take the latter unless he has perfect scores in all the 10 college level precalc exams that I have now downloaded and all from very reputable schools like Berkeley. I would like to sincerely thank UCBalumnus for pointing me in that direction.
Alexmer, your point is well taken about the level of difficulty of any math course he takes post AP Calc BC. We will definitely consider that and will talk to the school about that. My understanding is that the school has an MV Calc/Linear Algebra course so he cannot take a Honors level college course for that. But I will find out. It is more than 18 months away anyway if not 30 so I have time to do my research.
Skieurope, I read the article that you linked and I agree on most aspects such as not taking watered down courses and not encouraging taking courses with 19 year olds for 14 year olds. However the article doesn’t really tell me what to do instead of taking calculus and doesn’t provide an alternative list of courses to take. I do not believe not taking any math for 3 years in high school is an option as the school won’t allow that so would much appreciate your thoughts. The alternative is that he takes algebra and geometry again in high school and then preclac and calc but he would be bored silly in class then.
@LateCut - Here are some of my thoughts -
- It sounds like your son is quite gifted in math and I would advise him to take AP Calculus BC next year. However, I am also in little in your camp and I’m not as dismissive of the need to be “fluent” in pre-Calculus material as some of the other posters. He needs to know the properties of logarithms by heart, and a good math kid must know the double angle formulas, etc too. From your description it sounds like he doesn’t know logarithms by heart and that’s not good. But I don’t know if it’s worth having your son spend a whole year on it. If he enjoys math perhaps you could have him spend some time over the spring or summer working on this before Calculus in the fall. If it were me, I would have him learn the basics but not drill him to death on them. Instead, I’d have him apply the basics to problems that have some “bite” to them. American math curriculum books circa 2015 won’t have (m)any challenging exercises. It sounds like you might be Russian and have access to good texts; otherwise, there are some “old-school” American or English books from 50 years ago that will be at a more challenging level when it comes to trigonometry or even old-school “advanced” algebra topics like theory of equations, etc.
- I don’t know what your local college considers “Real Analysis”. At some colleges this is mostly measure theory, Lebesgue integration etc. I wouldn’t have him jump to that from his MV / Linear Algebra class (which is likely just to focus on computation). If it were me, there would be two possible next steps: 1) a class that is a good “Intro to analysis” course at the level of baby Rudin 2) a theoretical treatment of linear algebra at the level of Axler’s book. Choice 2 will be a bit easier than 1. A third choice is abstract algebra, but it wouldn’t be my first choice. Frankly, my advice is to choose based on who the most dynamic and inspiring professor is, not on the exact subject. Any one of these classes will give him a good sense of whether he'd like to pursue rigorous math, and the inspiration is much more important than what order he takes his classes in.
Or, he could take a more engineering math type class like differential equations if those are his interests.
- Have you looked into the online classes at the Art of Problem Solving website? There are some nice classes there as well as a very good online community focused on math competition stuff leading to the math Olympiad level. Lots of the Olympiad students take their classes, and I’d look into them. Competition math could be fun for him. Number theory is always something that gifted students his age find interesting – modulo arithmetic, quadratic reciprocity, etc. An advantage of this is that he can also learn some abstract algebra in a concrete setting.
- Spending a year on AP Statistics will probably be quite boring to your son. If it’s just a semester class at your high school then it will still be very easy but at least it will go faster. I’d be inclined to skip it or have him learn on his own at a slightly more advanced level when the time comes. It also sounds like he won’t learn much from AP CS. There are quite a few good online classes in more advanced computer science topics – check out Coursera’s algorithms course or MIT’s OCW.
It sounds like you are doing a great job helping guide your son and keeping it both fun and challenging.
Good luck to you both !!
@latecut I thought your last post was very insightful. The normal way most people do it is take after BC calculus MV differential and integral calculus and then follow it up with linear algebra. That is usually followed by ODE Stanford OHS is an excellent program that mimics their undergraduate classes. After that you can choose classes by time and professor that work into the schedule There are many possibilities. You can go from there to taking pde or financial math or number theory or combinatorics or if you have taken ap statistics you can take regression analysis,anova or time series. There is no one right answer. Your son has to experience some different types of math to determine what he likes. A lot of it depends on the time of the class and more importantly the person teaching the class. If you decide on Algebra the books of choice are usually Artin or Dummit and a class built around them. For analysis Stanford and UCB in large part use Ross. A lot of schools use Rudin. The best advice is to take a lot of different classes from a lot of different professors. You will hear that a lot . It is always better to take a class at a college than using something like AOPS. ( Learning to use office hours is also extremely important) Good luck
I was going to suggest Art of Problem Solving as a preparation for the national Math Olympiads, which your son would likely enjoy. Competition math will also place him in contact or in teams with kids who enjoy math at the same level he does and that would be important socially and developmentally (especially if you don’t want him to think of himself as “p”, which I apologize for using but did to emphasize we’re not talking about a kid ready to take precalculus in 10th grade, which is uncommon enough but not out of the norm and for whom a HS pace is fine.)
Neither my wife nor I have a very deep mathematics background so have not been able to offer much help to my son. He is in 6th grade and is currently taking Algebra II - he also recently qualified to take the AIME. I stumbled across Art of Problem Solving a few years ago and he has completed their Intro to Geometry and Intro to Number Theory books and is now working on the Intro to Counting and Probability book. My son seems to be a decent programmer in Python and is to learn Java this summer. I am familiar with the “calculus trap” and am unsure what we will do over the next few years (looks like potentially doing online Honors Precalc next year??) so it has been nice to follow this discussion. Every situation is different but from what I can glean from the discussion we are not doing anything too outrageous.
I do recommend the OP’s son to spend time on the Art of Problem Website, specifically Alcumus and For The Win - if he has not used this resource before, most of their courses can be self-studied but you will not get hs credit if that is a problem for you. My son enjoys working the problems.
Yearstogo, Your son is extremely talented if he qualified for AIME in 6th grade. Kudos to him and to you for being a great math dad! I wish him the best of luck and may be make it through USAJMO, MOSP, and finally IMO! You should be very very proud of him.
Florida26, I am not thinking as far ahead as college and using office hours. For now we have formulated a plan. First, he has to revise log and trig formulae which he did this morning and now remembers it all as he derived them all from first principle. But I know how things go with kids and unless he repeats it every day for a few months he wouldn’t remember it. Next he has to crack 100 in all the precalc exams that we collected. Then he can take AP Calc BC. Then we will regroup in another 3 years to decide what he takes in 12th grade as the sequence in 9th, 10th, and 11th will be AP Calc BC, MV Calc/Linear Algebra, Real Analysis/Linear Algebra.
MYOS1634 and Al2Simon, Point very well taken about Art of Problem Solving. I asked him if he knew about that and he told me that he has finished all the courses there except Olympiad Geometry and Group Theory which he will take over summer. He apparently also has been doing the AIME Problem series the past two years and is thinking about starting to do WOOT starting this Fall. I knew that he was taking the AMC and AIME exams through what the school organizes but I didn’t know that he was taking these courses too. If he is having fun taking them then all the better.
I would like to thank everyone who guided me in this matter and at this moment I think we have a good plan to pursue over summer and then be on autopilot for the next 3 years. In general we are very hands off when it comes to education so this is a good outcome for us. Thank you again for all your kind words and guidance.
@LateCut
I am slightly confused. If he has completed all of the AoPS courses (intermediate alg, pre-cal, and cal), he would be bored to tears in not only pre-cal, but cal as well. Did he actually take the classes online and/or complete the textbooks? If so, the only prep he would need for the BC exam is how to format the FRQ and understanding the exam format. AoPS courses are far superior in both depth and breadth than typical high school courses. (The post-tests for the courses would give you a fairly accurate depiction of the material he mastered in those courses.) I would not force him to repeat pre-cal for sure!
My ds took a similar path to the pre-cal in 9th path. He completed AoPS through intermediate alg in 8th and took pre-cal in 9th and cal in 10th. He took MV, diffEQ, and LA at a local university during high school (along with 5 cal up physics classes.) He is currently triple majoring in physics, math, and EE. He credits AoPS with his ease in physics and math. He says that the ability to derive equations is a huge blessing. (He enjoys proving every concept he encounters. He is rather weird that way. 
)
We have friends whose kids are even more gifted than our son who did not enroll their kids at a university for math after AoPS cal. Instead they followed the sequence with Stanford’s Online High School. I think someone else mentioned them in this thread. The courses are supposedly quite good.
FWIW, while RR writes about the calculus trap, he does NOT believe that advanced students need to held in a holding pattern. He is surrounded by kids who are incredibly talented and are mastering concepts at very young ages. I really love the talk he gave at Math Prize for Girls in 2009. http://mathprize.atfoundation.org/archive/2009/index
I know our son would have been bored to tears if he had been restricted to a slower sequence. We put the brakes on as it was b/c he would have spent way too much time on academics. But, we homeschool, so we could do whatever we wanted.
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I imagine his son has done the the introductory series with AoPS.