I was wondering whether anyone - 22B? others? - had anything to say about how to manage the situation in which a child has done all the maths - or any other subject, if similar issues arise - that's normally taught at school, but isn't ready for university yet. I'd be interested in plans, opinions, and, of course, experience.

DS9 isn't there yet, but it's foreseeable that he will be. In standard-US 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. *

Why not send him to university as soon as he runs out of school maths? DS is very asynchronous; it's impossible (now: could change I suppose) to imagine him being ready to enjoy life as a full-time university student a few years from now. He's anxious in new situations and fairly unskilled socially; exactly the wrong kind of person to take being a young university student in his stride. In the UK university's an all-or-nothing deal; you can't normally enroll in courses piecemeal. Some kind of special arrangement might conceivably be made with our local university, if that seemed appropriate, but (given that both parents work full time) the practical difficulty would probably make it impossible for him to actually attend.

Why not take a break from maths? Polarbear wrote:
I think you do have to keep going in maths, just as you have to keep reading, or exercising. If he weren't doing any maths I wouldn't feel he was being properly educated. Maybe it's just me, but it's standard for universities here to insist that students do take maths right up to when they leave school, even if they have the qualifications they need before then. Cambridge even discourages prospective maths students from taking a gap year. I think the latter's a bit excessive, personally, but yeah; let's take it that he does have to keep going in maths. In any case, there's no way he'd stop - the choice is between him doing maths with some kind of guidance and some kind of plan behind it, or him doing the maths he finds in a completely unsupported way. He may be ready for the latter by the time he's a teenager - so maybe this is a problem which will solve itself - but he isn't now.

Broadly I see the three kinds of maths available to him over the next few years as:

- the remaining bits of "school maths";

- university maths

- competition maths.

The questions are around what the balance and order of these things should be. Let me try to list the pros and cons of each as I see them.

School maths. Pro: close to age-appropriate. Possibility of taking exams to certify the achievement. Clearly laid-out dependencies, so easy to make sure you don't get into pedagogically bad situations. DH and I can, with a bit of effort, teach it all ourselves (though this isn't very important; he already doesn't need much in the way of teaching for material this easy). Con: not very challenging.

University maths: Pro: challenging, varied. Con: probably no possibility of getting certification of what he's learned so, for example, if he later wanted to go to university and read maths he might then be stuck redoing a course whose contents he actually already knew. Much more effort from us required to do things in the right order and put together appropriate materials, and he might need actual teaching, some of which we wouldn't be able to provide ourselves.

Competition maths. Pro: route to meeting other very good mathematicians. Challenging. Con: there's something very artificial about hard problems which are nevertheless designed to be soluble in a certain time using certain methods. Competition as such doesn't really seem to be DS's thing, and I could see him getting frustrated in a system where he was supposed to care a lot about winning.

What am I missing? Thoughts, comments?


*Aside: when and where do US students learn the stuff that's in our Mechanics modules - kinematics, rotation of rigid bodies, simple harmonic motion, all that stuff? Physics? University? It doesn't seem to fit into any US school maths course I know about.

Short version: nope.

Long version: there's the Open University, but its courses are not really suitable for a very mathy person. Anything else would be a special one-off arrangement. I know of one person, decades ago, who did have an arrangement that led to him getting a maths degree while still at school, but he had a parent available to drive him back and forth, and it was less flexible in various ways than what your DS had, I think. ETA a key element is that university here doesn't work by accumulating credits. There are courses you take in each year, and then you're done. Doing something different requires special permission and may or may not work practically. "Graduate school" is completely different from in the US; a PhD student here doesn't typically take taught courses at all, so graduate level courses don't exist in the same way, though much of the material to be found in US graduate courses is in undergraduate courses here and there are master's courses.

That's interesting, thanks for that pointer. (But ouch, on top of school fees!)

Edit to answer your edit: yes, there is plenty to be said for free learning. It may be that this will solve itself that way. Against it is the relatively inflexible university system he'd be going into if he read maths here later, as above. One way this could go wrong is if he'd taught himself half a maths degree, and then had to sit through all that again to get the other half. He says now that he'll probably want to read a science, in which case the issue won't arise, but he looks like such a mathematician in how he thinks that I won't be at all surprised if he changes his mind.

I will agree that Maths has a half-life.

Perhaps the online MIT courses may keep him stimulated and appropriately in condition when the time comes.

Two things popped into my head reading your post, although I don't know how feasible or appropriate they would be:

1. Can you talk to the head of the maths department at a local university and explain your DS's position to them and see what they might suggest? Maybe they have a solution that you haven't thought of.

2. Is it possible for your DS to take a break from maths at school but instead have a private tutor working with him in more obscure maths or with math applications in other subjects such as science or technology?

I can't really comment on the mechanics question -- not that familiar with the details, but I'm so glad you brought up the rest of this topic. It's something we think about often.

Even without the asynchrony you mention, I think there are many reasons to try to avoid early college. Every family and situation is different and I know early college works for many, but we are also facing running out of school math and yet (at the moment) hoping for college at a fairly typical age. I agree that university math can work but can also have issues. I think my son gets access to better thinking and more love of math in the math circle than in our local colleges. If local college options were better, that might not be true.

My kid hasn't found much "school math" to be useful and has consistently been frustrated, so we've tried to go outside the standard curriculum as much as possible. AoPS is an obvious solution and taking all those courses takes time. Despite hating repetition, I don't believe DS learned everything possible about geometry at 10, so Olympiad geometry, or more contest math, or more discrete math are all possibilities. I think repetition with more depth is very different that repetition of school type math, which he would find excruciating.

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.

I think it's been useful to focus on relative weaknesses in contest math, so speed and calculation have been drilled in math counts and been useful without increasing the level of math. I can see this frustrating a kid who really wanted to win, but it's been a radically different approach to math for my kid and useful, even though he likes thinking about a problem for days rather than one minute. While I think it focuses more on what I see as "tricks" than as deeper math, it's definitely been a learning experience for DS.

The wealth of online free options currently amazes me. If you aren't looking for credit but only growth, much of the math curriculum at some great universities is online. MIT OCW or coursera, udacity, etc. offer options.

We are trying to use the extra time to explore "mathy" topics more deeply that fall outside traditional math courses -- physics, economics, computer science -- can use math and apply it without being a typical math course.

A math circle has been huge for my son. IME, math professors running math circles are absolutely uninterested in age, or even in true readiness. I think my son entered a higher level circle before he was ready, but he was happy and engaged and so welcomed. He has grown to belong there. That exposure goes way beyond any curriculum and has fueled interest in math more than anything else. The people running it are dedicated professors who love math, love smart kids, and love feeding the fire.

We've also considered boarding school, but I'm quite ambivalent about it. Places like Exeter offer math far beyond typical high schools and I suspect such places exist in the UK??

I'm trying to focus on feeding the love of math and other subjects without worrying much about credit. DS reads a great deal rather than sitting through a formal course. He's self-taught a lot of programming in his free time. He's often frustrated with the formal course due to pace, so this has worked better for us. If you need credit, I think it gets harder.

We have already effectively run out of English, and of social studies.

Our answer has been threefold, but it DOES involve early Uni entry (then again, we're talking about a kid who is a 3+y skip across the board without any ONE area way ahead of others):

a) decelerate as much as possible until things are kind-of in synch (that is, drop into some kind of tolerable holding pattern with strengths until the most asynchronous areas come into focus too)

b) TUTOR at the levels the student has MASTERED. There is usually room for that kind of reinforcement that forces students with material/curriculum mastery to consider things in new ways as they explain it to others,

c) When the holding pattern is intolerable, or the option to revisit material via tutoring/explaining doesn't exist, independent study. (We're doing this right now with literature and social sciences.)


:nodding:

Yes. Exactly. Sociology and psychology along with non-traditional literature and writing, in my DD's case. We've also had her "detour" on her math track because of how the pre-calc and calc courses are taught at her school (they aren't, basically... they're canned instruction and all-assessments). That detour has involved tutoring lower-division math (pre-algebra through geometry and occasionally algebra II), and taking physics, stats and econ to keep skills sharp.

Finally, once we've run through those things, DD will be more than ready for college. In DD's case, she will begin college just after she turns 15, the summer after she graduates high school.

I wasn't sure about the timing of it all until about a year ago, though. I was frankly kind of TERRIFIED about 2.5y back, but DD has matured a lot more over the past year than I anticipated at that point in time.

We have not worried about credits here-- as Kaibab notes. My feeling is that if they aren't ready for a college environment, they probably aren't really ready to be racking up undergraduate credits on a transcript, either.


A gap year is another idea-- let them run through secondary and graduate, but then use the gap year as independent study time.

Have you read about the education of Terrance Tao? I ask because his parents found a very flexible path for him through an education system that was very (VERY) anti gifted ( elitism!!) and not flexible in this way at all.

This is a great thread. I'm tired now. I'll write something later.

Thanks everyone, lots of interesting thoughts there. A few responses:

- online courses: yes, it's great to be facing this now rather than a few decades ago. I'm not convinced that the Coursera etc. model is that great for maths, but at least it's an alternative to/supplement for books that gives greater choice.

- applications: yes, he's already doing some of this and will I'm sure do more.

- maths circles: yes, will watch out for such opportunities.

- talk to people at local university, check.

- boarding schools that go above and beyond; yes, there are some, and yes we're seriously considering them. The cost is a problem, and we're not sure whether sending DS away from two very mathy parents to get better maths at school makes sense; we'll see.

- credit: my only concern is, as I said, that university courses here are year-based not credit-based. If you turned up as a first year student having already mastered half the material in the first year, but not the other half, that would be quite a hard problem for the university to deal with. There are humans in the process, though, so maybe I shouldn't worry about it too much. And of course one can skip lectures :-)

- Terry Tao. Yes, I have read with great interest about his education. But do note that his mother gave up work to facilitate that solution. I can't do that and have him stay at his current school, and doing it and moving him into state education would be a backward step in all kinds of ways. Also, one article I read showed his timetable in one of those years, and he was actually getting a really weird mixture of whatever fitted - not a balanced education at all! Kudos to him, his parents and teachers for making it work for him, but it did have its downside.

I should have clarified my one point-- the thing that has made this work for DD is peer tutoring math(s).

So I didn't mean tutoring FOR your mathy kid-- by BY him.

This also can go a long way to improving social skills and confidence with older peers, too, which is ultimately a very good thing. DD has always had pretty darned good social skills, but they have definitely improved with tutoring. And she's found something that really lights her fire-- teaching.

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 1-1.5 years of college calculus) in 6th grade at age 11-12.

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 non-mathy 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.

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.


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 ;-) ;-)




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,
Unfortunately, "formally enrolled" is not further defined, though some countries (Canada turned up on my google) elucidate this as meaning enrolled on a degree-granting programme.

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 university-level 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".

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.

22B, I hope that you've seen Val's post in the other (pre-calculus 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

* self-pacing means that you can rip through the material at whatever rate seems appropriate

and the negative:

* it's the SAME (watered-down) math instruction from the same awful textbooks that B&M schools use

* there may be little-to-no 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 non-live 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 age-appropriate 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.

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 Pre-Calcuus 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.

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?

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 K-12 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.

I agree wholeheartedly.
How much do these cost for how long? What is there for elementary school kids?

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?

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.

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.

I came across the following, which makes me pessimistic. This guy was twice a USAMO winner, meaning he was in the top 12 meaning he could compete to get into the USA IMO team, but he didn't make the final 6. He was rejected from both Princeton and Harvard. (He was accepted by MIT.)
http://blog.tanyakhovanova.com/?p=150
http://www.maa.org/news/051209usamo.html
It's not clear why, but it's clear the selecting is not being done by the mathematicians, but instead by bureaucrats (admissions officers).

There's a financial issue here. Places like Princeton and Harvard (supposedly) select on merit, but charge fees based on financial means, as determined by the FAFSA formula. So a family earning say $75k/yr would pay $10k/yr instead of the sticker price of $60k/yr paid by people earning >$200k/yr. MIT (like most private and public universities) doesn't do this, so it is much more expensive.

Paradoxically the financially feasible options for us are Princeton/Harvard type places that "meet full financial need" or else the local state university (or delayed retirement). This forces us to set our sights that high, even though we really don't know that that's realistic.

I haven't checked out the situation with getting the appropriate credits for high school, but definitely will. And I am concerned that the school may be somewhat mediocre apart from the ability to accelerate.

But you have made the crucial point, that once you reach the high school courses you are locked into the standard pace. [For those who are unfamilar with Virtual Schools, the two main curriculum providers being Connections Academy and k12.com, the grade K-8 courses are much like home schooling, and you can accelerate through these courses, but in the grade 9-12 courses have scheduled compulsory teacher-led live online classes, and you must therefore move in lockstep with the rest of the class.] We were aware of this, and that is why we are having our son move quickly through the K-8 Math courses, because that's the only chance to accelerate (and they're really easy). Once he gets to the sequence Alg I, Geom, Alg II, Trig/Precalc, Calculus, he has to take the full year for each course, so he'll do them in grades 3-6. (Geometry and Algebra II can be done simultaneously, which is how these 5 courses can be done in 4 years, despite having to take the standard full year on each.)

You're lucky to have those courses in your district. Our school stops at Calculus (and it has Statistics, but not Differential Equations, Linear Algebra, or anything else). So our son was going to run out of courses no matter what, and we'd be having to find things elsewhere anyway, so there was no point holding back.

I agree with developing an interedisciplinary base, though I'm not sure how to do it.

I stand corrected. At least I hope I do because that's good news. I can't find where I thought I saw that MIT didn't meet full need. Actually the links you gave make it look like you'd pay EFC+$6k. Is that right?

I saw this, but in the past I've seen other things that I couldn't find again just now.
http://www.usnews.com/education/bes...s-that-claim-to-meet-full-financial-need
The big thing to watch out for is when they say they're giving you $X when they're actually loaning you $X.

You could have him progress through theoretical physics and learn and practice the needed math along the way. For example in electricity and magnetism taught at the level of Jackson, one learns about solving partial differential equations and special functions. Quantum mechanics uses that math and also linear algebra, and there are applications of group theory. Computational physics requires numerical analysis and programming skills.

Some philistines believe that applicability is what makes a branch of mathematics important or unimportant, so that following a math curriculum for physicists, engineers, economists etc. is a way of determining what to study.

22B, it's just me, probably...

but with a child who is just now, what-- six? seven?

I wouldn't be too concerned with choosing a college just yet. Two reasons for that.


1. You really don't know-- yet-- what that track of mathematics to calculus will look like. My DD seemed to be on that same track when she was 5-7yo. We figured that she might well be finished with calculus by 12yo. This was based on what I know now to be a pair of false assumptions on our part-- a) that pre-algebra on would involve the SAME level of cognitive demand and reviewing of material (it doesn't), and b) that she would continue to sail once she encountered genuine challenge in mathematics (she hasn't-- and the perfectionism set up in K-5 mathematics is partly WHY she hasn't, because she tends to regard being challenged as a sign that there is a problem, and that it is probably with HER). All of that to say that the child you have before you is VERY different from the one that you'll have at 14. We try to keep OPTIONS open, that's all. DD is reasonably well positioned to apply to Harvard or another Ivy, but doesn't want to.

2. You can't even BEGIN to predict what higher ed will be in ten years, what it will cost (not really, because things are probably reaching some tipping point), or for that matter what a "typical" or even "elite" path is going to look like. You also may not be able to predict what your own state will be demanding for a high school diploma, if it's anything like ours. The one thing to be cautious about is that you have enough math for FOUR YEARS of it to appear on a high school transcript somehow.
But-- as noted, I would not be concerned with this until your child is through with pre-algebra and maybe algebra I.


The other thing that I'm going to offer (both as another v-school parent and also as a STEM educator) is that this model is VERY poor in some respects, and high-school level STEM is at the very top of my list. The actual didactic instruction at the high school level is very much less than in a college setting, even-- and this is flatly not appropriate for most learners of any LOG. Basically, this means that your child may be in a position of learning calculus exclusively from whatever textbook is supplied (if there is one-- truly not kidding about that, btw), and from YouTube videos.

The biggest hurdle you may face in gaining admission to an elite (or any particular) college is in convincing the institution that the lab coursework taken in a "virtual" school is authentic enough to "count" as a prerequisite. Mostly, it's not, in my personal opinion.

But at any rate, all of that is a long way off. It's good to be thinking about things in a general way, but I've learned over the years not to plan more than a year or so into the future. I could not have really predicted my DD's current situation with math when she was 7yo.

I am new here and am thrilled to see this thread! My ds8 is exceptionally strong on the math front and we, too, have been trying to figure out what to do about math once instruction at his school runs out. Fortunately, they are keeping him moving several grades above where he formally is in school (2nd grade). Funny--the principal said to us with worry: "Eventually we will run out of instruction for him." We were just happy that they can keep things going! We told her that we understand completely that we will need to take responsibility (or a large part of it, anyway) for his math education well before the end of middle school (which is when his current school ends). So anyway, it is great to see all of these options laid out!

A couple of very informal ways that we've been "enriching" his math education so far:

1. Discussions with my business savvy brother, who helped my son "make up" a company, figure out stock prices, rates of return, etc.

2. Discussions of probability as offshoots of card and dice-based games.

22B--FWIW from my very anecdotal sense of what's happening with college admissions (based mostly on talented kids of friends): It does seem as if the top-notch universities care a great deal about well roundedness and also are attuned to "padding" with extracurricular activities rather than pursuing extracurricular activities due to true interest. As others have said, your child could change a lot in the next 10 years! Don't know if this rings true to you, but our philosophy has been to keep our DS interested and learning (and interested in learning)--and we assume that things will work out. Maybe that is delusional...

Ah, this is not so easy to explain. (And yeah, sorry about the length of that document! It was just that I wanted to give just one link, and that was the closest to comprehensive I could find.)

First thing to say is that we're very, very lucky with DS's school, I think (well, we chose it very carefully and we're paying for it, so that part's not luck, but we're lucky it's there to be chosen and that we have the means). They have people willing and able to do the right thing, and the resources to make it happen. So he's "going to his regular grade" but that doesn't mean he's doing the same maths they're doing; he hasn't done that since he was 5. His current maths teacher has a maths degree and has taught 18yos, and is rather ably giving DS his own work each lesson. Currently that's mostly problem solving - problems from the UKMT challenges and the like - but also, we keep in touch by email and sometimes I mention that DS could do with some exercises on [whatever] and his teacher will find some for him. A couple of years ago (when he had a different, less-mathy teacher before he got to the stage of the school where they have specialist teachers) he was working through a textbook that we provided in school, because that worked better in that context. It's very much play it by ear, without, actually, much of a plan at all - yet somehow, he learns, and always faster than I expect. < spooky music >

Well, except that partly I know how it happens - to feed DS's desire (which he does have) to have "new stuff" we also have books at home, of course, and we have an ALEKS subscription most of the time; he gets fed up with it sometimes and then we let it lapse, but he's wanted to go back to it repeatedly. Typically he'll do a few topics at the weekend, and more during the holidays. For example, he started ALEKS precalculus at the start of the Christmas holidays and finished it (bar the final assessment) in the Easter holidays. Yes, the ALEKS questions are mundane and I wouldn't approve if they were all someone was doing, but given that he's spending most of his school time (and of course some leisure time too) on problem solving, this works for us; he learns definitions and techniques in ALEKS (and I like that he's learning them by needing them in problems immediately, even if the problems are mundane; it's interesting to see how seldom he needs any explanation of how to do a "new" topic). So far, his getting to being able to apply them in harder problems seems to just happen.

I tend to be an over-thinker, but I think (despite having started this thread with my over-thinker hat on!) the thing we're doing right here is actually not to plan too much. If it turns out he didn't really get something in an ALEKS course and we didn't find out because he only got to use it on easy problems, well, no matter; he has time, he can learn it again when he's ready. So far, this doesn't happen much; in fact, the reverse happens - something that's happened more than once is that I watch him, make a list of things I think he could do with reinforcement of, and send them to his maths teacher, only to find that by the time his teacher gives him the reinforcement, he's mulled it over somehow and no longer has the weakness I'd identified.

Absolutely one can, in fact single subject degrees are the norm in England. (Actually, Scotland has a different tradition; where England has 3-year fully-specialised undergraduate degrees typically started at 18, Scotland has 4-year degrees typically started at 17 in which the first two years have some scope for taking courses outside your specialism. But there is never any requirement to take anything outside your specialism, as I understand is common in the US.)

Yes, in both systems, the last two years of school involve specialising, and the highest level maths qualifications are not designed to be accessible to everyone. Nevertheless, we still have "dumbing down" pressures and controversy about it and about what to do about it. Which I could write screeds on, but it's probably not of much interest, so I won't! I think the single best thing about the UK system is that school achievement is measured by externally set and marked exams, not by teacher opinion, fwiw.

I got an email today saying they've revamped Alcumus, actually, and haven't looked at it - DS goes in phases with it and hasn't been doing it just lately. So this is all based on how it was a few months ago.

They have prealgebra in it now; that's a fairly recent addition. Your DS would probably enjoy it now, I guess. I recommend it. Lots of interesting and competition problems, mixes up what it offers. At your DS's age, mine needed an adult to encourage/scaffold a bit, especially because (with the default settings, anyway) it will give you some easy problems and then throw in one that's really quite challenging. You can give up, but DS was never willing to! OTOH they didn't have prealgebra then, so that should help. Give it a go. You have to fax a form for under-13s, but they dealt with it very promptly.

We are a two-Arfken household, but all the same.... argh!

(Mind you, it would be cool if someone in the household actually read all of the three volumes of Feynman's lectures on physics we have lying around, I have to say. So yes, maybe.)

I agree, but I'm assuming 22B's planning in the same sense I am - knowing that the plan is not likely to survive the first encounter with the enemy, but finding that it eases anxiety to have some idea what the future might look like.

And of course, while a child who at 5-7 looks very advanced may look less so later, it can go the other way, too. A plan that has courses in lock-step with a one-course-a-year plan once a child reaches high school level makes my blood run cold - no way would that have been feasible for DS. In the spirit of calming anxiety, I've had a top-secret :-) "what he might be doing when" document for years. I revise it regularly, but I'm still waiting for the time I have to revise it because he's going slower than I'd planned.

Thanks, everyone for all the answers, ideas, suggestions.

It's good to hear Alcumus now starts at Prealgebra, instead of at Algebra. That is the right level for DS7 so we can start it now. (He's only "offically" up to grade 5 maths, but I think he has "unoffically" learnt most of Prealgebra just messing around by himself on the internet.)

As to the AoPS online courses, I think he would need an adult with him, watching and typing. Has anyone done it this way? Could it still move too fast? I'd be interested in seeing AoPS's approach to these competitions anyway, as I'm not familiar with the American competitions.

ColinsMum, could you recommend some sources of problems in addition to Alcumus. We've almost never had DS7 solving (non-routine) problems at all, yet, as we were just getting him through the basic K-5 maths so he'd have some basic knowledge, but now he desperately needs to be challenged.

Another question, what's your approach to assessing "mathematical maturity" and readiness for certain mathematical activities? For example, I haven't mentioned at all to DS7 about theorems and proofs, since he's not ready for that. He can understand and explain things, but I'm just happy for him to think about things without worrying about rigor at this time. Know what I mean? What other stages of "mathematical maturity" should I be thinking about?

Another question, anyone know of a good resource (especially online) for learning very basic logic (and, or, not, quantifiers) and the same for set theory. These topics are totally absent from the school curriculum, so this void needs to be filled.

HowlerKarma, I certainly share your concerns about Virtual Schools (and thanks for your various warnings). At least they allow significant multi-grade acceleration, although these are just courses for average students.

As far as planning, we try to make it robust against unforseen changes. The timeline is a reasonable estimate, but it could change a bit. Actually, once the high school courses are reached, and progress is (mostly) locked in at one course per year, then that'll be the time to branch out into many supplemental activites such as AoPS. This thread should be about not only what you do when you run out of school-provided maths, but also what extra things you do before you run out.

As for college "planning" obviously we don't know what options our son will have, and that uncertainty is not really an issue academically. But financially (in the US) it's a big deal. With our modestly above average income, the best universities are the cheapest, since they offer the most need base aid (with the possible exception of state universities within ones state, which may be cheap but not good). I did a comparison of costs at Princeton and MIT using their online financial aid calculators
https://npc.collegeboard.org/student/app/mit
https://swebapps.princeton.edu/FinAid/finaid_form.pl
and found that, with our numbers, MIT cost $11k/yr more, and most places would cost more still.

Books to prepare for competitions such as Math Olympiad, MathCounts, and AMC, for example

Math Olympiad Contest Problems for Elementary and Middle Schools, Vol. 1
by George Lenchner

are sources of challenging problems.

This is a good post. I often wonder what will happen with my DS6. He is in grade one, doing grade four maths (including grade five pre-algebra) and grade three English , but only grade two in everything else. Going at this rate, he will finish his maths studies when he is 13. Universities here don't allowed under 16's on the grounds, but you can go through open universities, so I guess he could do that for fun. We'll see how things go.

The topic of college costs has come up here (and in other threads) so I thought that topic could be split off from here and consolidated in a new thread.

I started a new thread here
How much does college cost? Unified thread.
http://giftedissues.davidsongifted.org/BB/ubbthreads.php/topics/159046.html

Bostonian, thanks for the suggestions.

squishys, how is your son doing that. Is he actually in classes with older children, or is he learning it elsewhere (or a combination)? Some have siad that there are obstacles to acceleration in Australia, but you seem to have managed some significant acceleration.

I believe your kids will be able to fly through the AoPS Pre Algebra in no time.
My son finished 5th grade math in 3rd grade. He just finished 4th grade this year with the regular curriculum (probably not a good idea). We also went through the AoPS Pre Algebra book and videos this year. We also had a tutor once a week for general Q&A if needed.

He also got his feet wet on competitions. He LOVED it. Kids like him, where have they been? You could not wipe the grin off his face.

I also stuck a couple questions from the AMC8 on his school folder several times a week. Just for something to think about. these are great questions. They seem to have a range of questions for every level.

thanks for starting this thread.

22B, my son sees a private tutor for maths and English, and everything else is done as an extension through school. Next year his school will grade skip to year three, and that is the grade when the school starts taking giftedness seriously. So I'm hoping, since my son is getting 100% on his work with the tutor, that they will give him extension in maths. If not, my husband and I decided part time homeschooling for maths and English- so either way it's happening.

Australia seems to be one of the worst for helping gifted kids.

(The tutor teaches according to the curriculum and also the same methods, etc, to keep my son in line with the school and my son's teacher, as the school already teaches a little ahead compared to other schools in the state)

Also, I have already found my son a high school. It had a gifted program ( a rarity in my state) where you can consolidate some grades, so instead of taking five years to complete it can take three years; then the child can take early uni classes to shorten the degree once at uni.

I know it's early times, but I couldn't imagine my son losing his maths abilities. He has had them since he was a baby, and they're only getting incredibly stronger.

Squishys, if he's really ready then there will be a way, Flinders Uni has made it work at least once.

Really? I was looking at age requirements and the earliest I found wad 14, but in a different state. If Flinders also allows it, that's great. Although, I don't know if I would I ready for it! Uni is so grown up. That's why I thought with OUA he would be in the safety of his home. But, it's a long way off. I'll reevaluate in a few years.

squishys, in case you didn't catch MumOfThree's reference:
http://en.wikipedia.org/wiki/Terence_Tao

I don't catch on to anything lol. Thanks for the link!

Hi Squishys, UNSW has early admission for students 15 years or younger. http://www.unsw.edu.au/future-students/early-admission-exceptionally-talented-students-scheme. My wife works there and we have DS4 who is EG so it has come across our radar. Just need to get through the next decade!!

I'm not sure if I could send my baby away to another state any younger than 18! My son has an ambitious goal of finishing uni at 11 LOL!!

Squishys my child is not a math prodigy in even the slightest fashion, so I have no experience here. But honestly I think that these kids of ours are so unpredictable that you dont want to panic too much yet. And even in our anti gifted state, truly out there kids have been accommodated successfully before, so you know that handled well, for the right child, a path can be found...

I'm so naive about this gifted stuff it's embarrassing. I've only just learnt that you can be verbally OR mathematically gifted, or both, in the last year *blush* I am a very organised person so I like to have a plan. I am following my son's lead, so it is good to know that there are options. I really don't think, though, that I would send him on campus, just online uni stuff for fun.

It is amazing how anti gifted this state is. Although I find it is just the school system and teachers- the people are quite open and excited to hear about gifted stories, in my experience.

Squishys I would tread carefully with who you talk to socially as well to be honest, I am extremely cautious these days. I've actually found (with a psych report in my hand) that schools have not been too hard to deal with, they cant always help, but they don't deny what's on the paper in front of them. Have you looked at PAC and Saints? It's where I would start looking if I had a boy, particularly an asynchronous mathy boy.

I don't share info too freely, but some people notice. By 'people' I meant more like my friends and family.

I have considered PAC, but I'll probably go with Glenunga International (since we are atheists). It's a toss up between the rich kids or the smart kids. I have no idea. I'll start researching it more in a few years.

I've been following this thread and wanted to thank all the posters who've contributed to the discussion. This is one of the most informative threads I've seen on the forum!

Logic puzzle books by Raymond M. Smullyan have been widely popular among 'mathy' children for decades. Try:
- "What Is the Name of This Book?"
- "The Lady or the Tiger?".

Smullyan, 94, is "an American mathematician, concert pianist, logician, Taoist philosopher, and magician". Here is his picture and his story (in his own words, lower on the page): http://mysite.verizon.net/vzeeaya7/raymondsmullyan/ .

Also consider books by "fellow polymath" Martin Gardner.

Pre/post-tests (online) for AOPS books and courses would be useful for this.


AOPS "Prealgebra" book asks students to construct proofs (AFAIR, distributive property, etc.) at the very beginning.

kcab and arlen1, thanks for the references.

Has anyone tried eIMACS?

-------------------------------------

To the Australians, this may be of interest
http://www.amt.edu.au/
http://www.amt.edu.au/mcya.html
http://www.amt.edu.au/amcfact.html

Thanks for that, 22B. I think my son will be very interested.

Yes -- for the computer science U course. It was ok, but DS found it tedious and repetitive. There were lots of "exercises" to do.

Good to know, kaibab.

[back from a week away and catching up]

Topically, here's an article about Alex Thorne, who has just finished a master's degree in maths in parallel with being at school for other subjects. Depressingly, "His mother [...] had to sit in on the lectures with him because of his age", i.e., it's yet another case where the solution only worked because his mother was available during the day to make it work, i.e. didn't have a typical career or gave it up.

You've had some good answers while I was away, but here's mine anyway. We've mostly used past papers from maths competitions. Early on the ones which are multiple choice worked much better than anything where he had to "show his working"; that's less of an issue now, and ymmv anyway. The three families of UK contests we've used are:

The Mathematical Association's Primary Maths Challenge, aimed at children up to the age of 11. A few papers are available here (February is the follow-on round, so harder than November) and there are also books, usefully sorted into challenge levels, e.g. this.

The UKMT ones - very usefully wide range, starting from the Junior Maths Challenge, aimed at children up to 13. One of each paper available from their website here, and again you can buy collections. The mentoring scheme papers, here, are also worth knowing about (perhaps for the future more than now, or if not I'm seriously impressed!)

The Scottish Mathematical Council ones, see here. In the actual competitions, these require full written solutions.

School have used scholarship papers (for entry to senior schools at 13, e.g. these or these) too. Unfortunately the ones that are on the web are the exception.

Interesting question, not least because it isn't something I've been in the habit of thinking about at all!

I suppose I suck it and see. I can think of three occasions when he's worked at something that, with hindsight, he wasn't ready for:

- Probably the first time I ever saw him have what looked like conceptual difficulty with something was when, at 7, he came to a section on cumulative frequency graphs in the textbook he was working through in class. He had trouble getting what it was they were showing and how. He could do the questions mechanically, but didn't like it and made weird mistakes. I helped him through the material in the book's (short) section as best I could, and didn't linger, because it really felt like something he wasn't ready to learn yet, rather than like something where he was having trouble that would be productive to work on. This - and especially the fact that I don't really understand what the problem was (given that he has had no problem with any other kind of graph or data handling) - is probably why I've been holding off on him doing more stats, actually.

- That same year he did the (now defunct) ALEKS course on high school chemistry. He'd been desperate to do this for a while, and I'd insisted on his getting at least to the end of ALEKS's grade 6 maths before he did so. I hadn't realised, though, how far the course would take him, or I might have tried harder to put him off starting it! It started fully appropriate, but got much harder towards the end. I did suggest fairly forcefully that he stop and come back when he was older. He insisted on finishing the course, but there was a lot that didn't stick. (He's now doing their introductory college chemistry, which has a lot of the same material, and it's a different story entirely.)

- The other thing he'd been very keen on learning for years before I let him at it was calculus. (I had a short thread here, actually.) I subscribed to the eIMACS course and put it on the list of things I let him pick from to do on the bus (we have a long bus journey together to school in the mornings and that's where he gets most of his mathematical input from me!) He started it with great enthusiasm and did a chapter on limits (that was useful, I think) and one or two chapters on differentiation, but just gradually asked for it less and less often. The year's subscription lapsed without his getting to integration and I didn't renew it. In this case, it wasn't really that anything was hard - it was more that he'd satisfied his curiosity, perhaps. I didn't attempt to keep him interested, because I thought it'd go better after he had more trigonometry, anyway.

[ETA and to answer the later question about eIMACS, I thought this course was OK, but not great. Understandably, not many of the exercises were automatically graded, so he definitely needed someone there to check he was understanding; well, there are answers available, but since other correct answers are possible, that's of limited use. Pedagogically it was not bad, but there was a fair bit of "in the AP exam you can expect" to be ignored.]

I suppose my general attitude is that he has time, and it's fine to be guided by his interest. If he isn't ready for something, he can leave it for later, no problem, or if he wants to have a go anyway but it doesn't stick, he can do it again later, equally no problem.

With DS I wouldn't, incidentally, have picked proofs as an area to delay for mathematical maturity reasons - he's been very interested in proof, and proof theory, for a long time (there's a resource I would love to recommend, but can't till it's finished and published!). Writing proofs longhand has been another matter as until recently handwriting was a problem; I did have to push back against a teacher who wanted him to focus strongly on "showing his working" for this reason.

As kcab said, Language Proof and Logic - for the software (more than just Tarski's World here, though that's good), more than for the book, which obviously isn't written to be appealing to children and isn't particularly so to DS. And this other resource that I can't point you at yet!

HTH!

PS the other resource I'd strongly advise considering is Descartes' Cove. It's not cheap, the graphics are dated and there are mistakes (mostly where the communication between the graphics people and the maths people went wrong so diagrams intended to be explanatory are misleading) but DS loved it and learned a lot from it.

ColinsMum, Thanks for all the sources of problems. That's very useful. We really haven't tried this, but we need to. DS7 does edutain himself with lots of maths stuff on the internet, but he hasn't really been exposed to a lot of challenging problems.

As for "mathematical maturity", I guess your right that it's okay to have some false starts when they're way ahead anyway, although I wouldn't want a false start to happen in his school courses. If a competition or AoPS course doesn't work out, it's not the end of the world. Better to try, as long as it's not so traumatic that it instills fear in trying very difficult things (aka "perfectionism").



FWIW I think software can be pretty sophisticated at recognizing equivalent answers.


Is the software appealing to 7yo children. I'm looking for the most extremely elementary parts of set theory and logic (and maybe graph theory) just to be aware that these topics exist at all, because they're just not in his courses at all.

FWIW, I think perfectionism is more easily instilled (was in me!) by never trying things that are too hard. But you the parent have to really, really believe it's OK to try and not succeed yet, and I take the point that this is lower stakes at home than at school (though I don't have virtual school experience and that must be interestingly different).

Can be, sure - that's why I wrote "understandably" not "inevitably". It's still hard work to make it that sophisticated, though.

It was to ours - not wildly so, but enough that it saw a fair bit of use over a year or so - but of course ymmv. He used to like, for example, making a set of sentences in this interface and challenging me to build a world that satisfied them all, and vice versa.

Have you considered math competitions? Even if DS doesn't want to compete, you may be able to find out about math enrichment opportunities in the UK. Try contacting someone at the UK Mathematics Trust: http://www.mathcomp.leeds.ac.uk/about-us/.

Another option is Art of Problem Solving. In addition to courses, they have communities that might keep DS active in interesting math until he's ready for University: http://www.artofproblemsolving.com/

Yes! He's doing very nicely at these, thank you :-)

Also already in the mix. He's actually a couple of weeks into his first taught AOPS course now, and I'll post more about that when we both have a better idea about how it's going.

Thanks anyway :-)

In another thread
http://giftedissues.davidsongifted....Re_Ivy_League_Admissions.html#Post162766
this link was posted
http://www.oregonlive.com/news/index.ssf/2009/03/eugene_high_school_student_win.html
Here "freshman" means 1st year high school -- 9th grade.
So that's one data point that someone can take university maths courses and still go in the IMO.
http://www.imo-official.org/participant_r.aspx?id=15882

DS finished his Extension 1 and 2 Higher School Certificate maths in Year 9 and it wasn't going to be an option that he go three years with no maths.

I spoke with the school and they acknowledged that it wasn't a suitable outcome so the Principal approached the local university and arrangements were made for him to enrol as a external student in a BSc. The agreement was he would concentrate on completing only mathematics units initially and as time progressed branch out as he desired.

He started with one unit per semester and attended residential schools where appicable.

As he completed further school subjects ahead of time he picked up more university units. By this time he was driving so he pretty much spent the last two years of school splitting his time between school and attending lectures on campus.

His enrolment at university did not affect his entering mathematics competitions here although he did lose interest in them so rarely entered.

That's great to know, matmum! Australia seems a bit behind the mark when it comes to gifted education... Your son is lucky to have a supportive school. May I ask which state you are in?

Hey squishys,

We are in Northern NSW. My son attends UNE (University of New England). He was interviewed in 2008 and commenced in 2009. There was also another lad there from QLD doing a BSc majoring in Mathematics and he was 12y/o.

I'm in SA. That is great I hope my son's future high school will be as supportive (should it ever come to that).

We live in a small rural town so DS attended the local central school (public). It is K-12 with a middle school and I think that was the clincher to them being able to accommodate to his needs and those of his sister.

It wasn't easy for the school but they were happy to subject accelerate and timetable concurrently to achieve it. DD and DS chose subject acceleration over grade skipping.