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Joined: Feb 2013
Posts: 1,228
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We are in a very similar situation with our 7yo son, in some respects. He's specifically much much better in mathematics than his other subjects, and he clearly just naturally thinks like a mathematician. He's in 1st grade just finishing 5th grade math, and we anticipate him finishing AP Calculus BC (covering 11.5 years of college calculus) in 6th grade at age 1112.
Since we are in a (free public) virtual school, he can go at any pace. Basically we are homeschooling using the "canned" courses from k12.com, which are designed for average students but which are designed to be compressible, to cover material quickly for the strong student, so he can go through all the material without gaps.
We have 2 Maths PhDs, but 1 income (by choice) and 3 kids (2 not yet in school), so we are very much focused on bringing up, and especially educating, our children ourselves directly, but have to minimize costs. The schools are mostly really awful here, in various ways, so we have few options.
Yes, of course one has to keep going in maths. We don't have a detailed plan, just some rough ideas. We'll supplement with AoPS courses, for all topics not on the standard path to calculus, and the competition preparation courses, whenever he is ready. (The few hundred dollars per course is an okay expense. Question: are there any other "schools" of AoPS's ilk that are worth looking into?) There is a school in the state (but far away) that has more advanced, or different, courses such as multivariable calculus, differential equations, discrete mathematics, etc., and these courses are available for free (within the state), so that will cover a couple of years maybe. After that, there is the possibility of courses, or maybe reading courses, and the local university. It is a fairly average state university, but a lot of the faculty have PhDs from Ivy League/Oxbridge type places, so there is plenty that a smart schoolkid can learn from them. And we can plain old fashioned homeschool using books and our own mathematical knowledge. Maybe he could do some research. There are logistics to work out with all this. It's just a vague plan. One concern is I was wondering if taking university courses (while officially being a seconday school student) could disqualify you from competing in certain maths competitions. We are not going to worry too much about credit, as long as university entrance (and high school graduation) conditions are satisfied, and as long as universities at least somehow take into account all the "extracurricular" maths.
As for maths competitions, whatever their drawbacks, I think one has to compete. They are a way to see how one measures up against others in your region or country (or the world). There's competition to get into universities and to get jobs, so competition can't be avoided. And a string of very good competition results may be regarded as more signicant than rapid progress and high marks in easy schoolwork. These competitions (or sequences of competitions) really can identify people as being not just top 1% or top 0.1%, but even top 0.01% or rarer, and that kind of identification can help.
When I was a kid, I never heard of people preparing for maths competitions. I just thought they were fun, and did well. But I see these days there are competition preparation courses and math clubs/circles. While "teaching to the test" would be a sad thing, I gather these preparation activities are just a good way to learn some mathematics that's not in school, and to interact with similar kids, so that's definitely something we'll look into.
So after finishing calculus in 6th grade, our son won't be twiddling his thumbs for the following 6 years waiting for the next maths course to show up at uni. There's plenty of maths he can do in the meantime, even if it takes some scrambling and improvising. If he has to repeat some material, hopefully it's at an elite (Ivy League/Oxbridge type) university, where it's presented at a much higher level. (I was looking at the Princeton University website once where it said words to the effect, my paraphrasing, "yeah, sure, you mighta taken calculus before, but you haven't taken our calculus", and they do have a point.) We don't yet know if he'll be good enough for those places, but maybe competitions over the next few years will give us a rough idea where he stands.
As to the issue of whether to start university early, here's why not for us. Our son is fairly good at all his subjects, but he is absolutely not the kind of kid (in contrast to some on this forum) who could accelerate multiple years in all subjects. (Actually he's 1 year accelerated across the board, so he could conceivably start uni at 17 instead of 18.) Instead he's specifically very good at maths and less good at the nonmathy subjects, so he'll probably continue those at the regular pace. And there's the usual considerations such as maturity, social eptness etcetera. But another consideration for maths is that when it comes to competing for entrance at an elite institution, it's very hard for a 15 or 16 year old to compete with a, say, top 0.01% 18 year old, which is what I'm guessing it takes to get into these places, though I could be totally wrong about that.
By the way, does anyone know what it takes to get into maths at an elite institution? Is it based purely on merit? Or do you, as some have suggested on this forum, have to fluff your CV with extracurricular activities like volunteering at the homeless cat shelter and playing polo?
One final thought. There are a lot of jobs where mathematical ability is important, but very few jobs as a research mathematician. So you have to have your child prepared for this uncertainty.
I'm sure I've forgotten to say several things, but I need to sleep now.




Joined: Sep 2008
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Question: are there any other "schools" of AoPS's ilk that are worth looking into? Nothing seriously comparable that I've managed to find, and I have been looking. Many less good things. I've been watching what's provided by the DaVinci group (organisation has been hopping around with funding, current page here) but haven't joined/used it yet. I see they have added some maths provision, OxMaths, since I last looked, but it's not suitable for your kid or mine. Maybe he could do some research. Ah yes, the "grow your own collaborator" plan. DS wants to prove Goldbach's conjecture; we'd far rather he proved P ne NP (and not only for financial reasons), but we'd settle for Goldbach if that's what lights his fire ;) ;) One concern is I was wondering if taking university courses (while officially being a seconday school student) could disqualify you from competing in certain maths competitions. This is a valid concern, I think, and the rules are quite likely to change given the fluidity of the current situation, so it's one to watch. For the IMO at present, Contestants must not have formally enrolled at a university or any other equivalent postsecondary institution, and they must have been born less than twenty years before the day of the second Contest paper. Unfortunately, "formally enrolled" is not further defined, though some countries (Canada turned up on my google) elucidate this as meaning enrolled on a degreegranting programme. So after finishing calculus in 6th grade, our son won't be twiddling his thumbs for the following 6 years waiting for the next maths course to show up at uni. There's plenty of maths he can do in the meantime, even if it takes some scrambling and improvising. If he has to repeat some material, hopefully it's at an elite (Ivy League/Oxbridge type) university, where it's presented at a much higher level. (I was looking at the Princeton University website once where it said words to the effect, my paraphrasing, "yeah, sure, you mighta taken calculus before, but you haven't taken our calculus", and they do have a point.) They certainly do (and if you were talking about Oxbridge literally, the course assumes you've done plenty of calculus anyway, since it's on the normal school syllabus here rather than being nominally university maths). All the same, if much of his six years after Calculus BC turns out to be spent doing universitylevel analysis courses and research in that field, he could still easily end up more suited to teaching Princeton's intro calculus course than taking it... but here we surely come to "plans are useless, planning is vital". By the way, does anyone know what it takes to get into maths at an elite institution? Is it based purely on merit? Or do you, as some have suggested on this forum, have to fluff your CV with extracurricular activities like volunteering at the homeless cat shelter and playing polo? It will surely depend on which elite institution, but I can say for sure that neither Oxford nor Cambridge could care less about anything but academic merit, because they're both on record saying this clearly. I sort of doubt that someone who had IMO medals and/or papers in reputable journals to their name, and didn't have two heads, would in practice get turned down even at US elite institutions  but it would be good to hear from someone who knows.
Email: my username, followed by 2, at google's mail




Joined: Feb 2011
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22B, I hope that you've seen Val's post in the other (precalculus textbook request) thread: secondary math and textbooks and pedagogy, oh my...This has very definitely been our experience. I'm sure that you won't overlook it, given your background and the fact that a parent is home with your son we certainly didn't miss it, that's for sure (we're with Connections). The pluses of such online programs: * you go through the ENTIRE textbook each year including those ending chapters that B&M schools usually skip * selfpacing means that you can rip through the material at whatever rate seems appropriate and the negative: * it's the SAME (watereddown) math instruction from the same awful textbooks that B&M schools use * there may be littletono actual instruction from a live teacher for more advanced mathematics, which is only okay for true autodidacts. When DD was seven, I'd have predicted her to be in calculus last year, too (that would have been when she was 12). Didn't happen, and I'm glad. I do think that you're right to be considering what to do when he runs out of math... because the asynchrony is going to be a real bear... but my personal opinion (our DD has two parents with PhD's in the physical sciences, btw, and she's a rising HS senior as of the end of next week) is that primary and secondary mathematics teaching/pedagogy is weak and getting worse by the minute. I absolutely would begin making a plan to supplement with authentic materials. Depending on the type of learner he is, maybe Great Courses has something he'd like, too. We've used a few of their things, but DD's learning style isn't terribly compatible with nonlive instructional methods. I also hope that your DS continues to tolerate the pacing/level of 'instruction' via K12. My DD has NOT tolerated it very well. It's been a continuous battle for over 7 years. I was seriously angry over the gutting of geometry, and so was my DH. It ruined that course. Ruined it completely for kids with the math ability to fall in love with it. Gaaa. Oh and the other thing to watch for since we're using that same virtual schooling model? Make sure that he can continue to work at his own pace in secondary. That's a huge catch with Connections. They can't; they MUST work synchronously and in order once they reach secondary math. Also make sure that if you're going to venture outside that system for enrichment/alternatives, that you've satisfied the requirements for graduation and have the requisite coursework listed on a high school transcript somehow. This may mean that your DS has to take "high school" geometry when he's 9 which also means that any ageappropriate flakiness has lasting consequences. If they tell you that you can use local university credits to substitute for AP Calculus get it in WRITING. We've found that national is surprisingly (or not, perhaps) stubborn about "you should take OUR class... we offer Calculus/Chemistry/Econ/Psychology" Yeah, but your version is a canned JOKE... and I want my DD's first experience with this subject to be, you know... authentic. "We offer that class." Just noting that. BTDT. My DD has had to take some really worthless electives. Anyway. I mention all of this because it was absolutely NOT obvious to me when my DD was in primary grades just how awful the secondary math instruction has become. If K12 is anything like CA, they also won't let you do much "previewing" of course syllabi, either, nor of content. I mean, it's great to have a parent to offer direct instruction when that is a major deficit in a program (we have that problem here), but it only works when there is some real content within the course. Otherwise, you wind up shooting them in the foot because the assessments are aimed at something totally different than the level that they understand. Don't even get me started on "front end estimation" and ALL multiple choice assessments in this model.
Schrödinger's cat walks into a bar. And doesn't.




Joined: Feb 2011
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I don't have quite the same issue but I have looked ahead and made decisions based on the distant future. I have less of a problem because DS is not a math prodigy and is also equally strong verbally. We chose to wait to do Algebra next year (5th grade) even though DS appeared ready by every measure. This way he won't start PreCalcuus until 8th grade, which will leave him enough math in high school  Calculus, Differential Equations, Linear Algebra and Statistics (current offerings in our district). There is also a lot of math horizontally. DS has picked up odds and ends by reading interesting math books (not textbooks). DS has expressed some interest in business math and econometrics. My thought is to help him develop an interedisciplinary base, which is actually more beneficial in the long run. Of couse, he won't be ready for something like econometrics until he has mastered Calculus and Statistics. He will also likely do some competition math.




Joined: Feb 2013
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After finishing the usual highschool sequence, my son took the undergraduate math courses at the university, but as a highschool student. When he matriculated and started as a freshman at his current university he started taking graduate level courses. He has never had to repeat anything. He has also self studied topics and has been allowed to skip taking the courses formally by showing mastery through discussions with the professors and, in two instances, taking the course final exams. ( didn't receive credit, just was able to waive them as prerequisites to higher level courses) What (kind of) universities were these where could take undergrad during high school and grad during undergrad? Do you think you were very lucky to have no forced repetition and to get credit for courses taken, or do you think this is to be expected?




Joined: Feb 2013
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DS9 isn't there yet, but it's foreseeable that he will be. In standardUS terms, he has most of AP Calculus and most of AP Statistics still to go. In UK terms we have a bit more flexibility, because there are more options in the final two years of school maths than any one student normally takes; if we have him do it all (and whether this is sensible is one of the questions in my mind), it'll keep him going for a few more years. Here's one fairly typical syllabus document. * I had a quick look at that 157 page PDF document. Obviously your son should just do the whole lot if possible. From your comments in various threads I wasn't quite sure how he's covering this material, since he's just going to his regular grade in a B&M school. How is he doing it? Yes "Mechanics" is part of Physics in the USA. Also I see the subject area called "Decision Mathematics" which looks more like Discrete Mathematics. That's an area (if interested) that he could go a lot further in without clashing too much with the university courses (since the area is somewhat neglected in many departments). The UK K12 syllabus certainly covers more than in the USA. I assume that's due to earlier specialization, and due to not lowering the level so that more people can reach it. It's true that one can do 100% maths in a UK undergraduate degree, right? American undergraduate degrees are far too broad, meaning not enough maths gets covered. Anyway, that's an argument for covering material early, just to get to a reasonable level. Someone was questioning in another thread, why would anyone bother to get a PhD, just to end up teaching high school math. Well the answer is that you need a PhD to get a job at a university.




Joined: Feb 2013
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Thanks everyone, lots of interesting thoughts there. I agree wholeheartedly. Math summer programs in the US usually cover math outside the traditional curriculum (Mathpath for middle schoolers or Awesome math, or for high school, Promys, HCSSiM, etc.). These are expensive but also many are international and provide access to higher math and math growth in the summer at least. How much do these cost for how long? What is there for elementary school kids?




Joined: Feb 2013
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Question: are there any other "schools" of AoPS's ilk that are worth looking into? Nothing seriously comparable that I've managed to find, and I have been looking. Many less good things. A general question to anybody: does anyone here who has actually used these course have some feedback about them? Apparently lessons are live online, but all communication is by typing into a chat box with no sound or video, not sure about pictures. How suitable is this format for elementary school kids? What level does Alcumus start at, and what is that like?




Joined: Feb 2013
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Maybe he could do some research. Ah yes, the "grow your own collaborator" plan. DS wants to prove Goldbach's conjecture; we'd far rather he proved P ne NP (and not only for financial reasons), but we'd settle for Goldbach if that's what lights his fire ;) ;) Okay "grow your own collaborator" is very funny. Actually "grow your own scribe" would better improve my output. (But we're not growing a scribe.) If it comes to financial reasons, P=NP is more lucrative. Seriously, "research" can just be a toy research project to dip one's toe in the water (depending on one's level). It's just another activity outside of regular school maths.




Joined: Feb 2013
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..."formally enrolled"... Okay, I admit having peeked at this regulation, though there's 99.x% chance we won't need to know its exact meaning. But if you ever find out, let us know, just in case.




