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I have a 9-year-old who really loves math. He seems to have taught himself through about high school Geometry by reading library books. His 8th grade brother who just started Algebra 2 thinks the younger one knows about as much math as him. I accept that it's not reasonable to expect his school to teach him anything in math (he's in a 2-years-ahead gifted program and all the other subjects suit him fine), but I asked him if he would like me to give him stuff to work on at home, and he was really excited about the idea.
Does anyone have any materials or strategies to suggest? We tried Art of Problem Solving pre-algebra briefly last year (like maybe 4 or 5 evenings), but he's not excited by math contest type problems and the concepts are not presented in a very interesting way. He likes things like graph theory, complex numbers, formal logic, game theory, different dimensions, probability, topology, etc., which he read about in library books. For fun he does things like figuring out how to make logic gates with dominoes and finding the volume of polyhedra. He wants to learn calculus and prove Goldbach's conjecture.
At the same time, I know he has some holes in his knowledge because he didn't learn anything formally. Like, a few days ago, we discovered he can explain to you what the quadratic formula and factoring are for but he didn't know how to factor (we told him and he was doing it in his head in 5 minutes). (He does have all the basics like exponent rules down though.) So I would like to teach him more systematically. But still keep it interesting and fun. If there's anything that doesn't involve screens that would be preferred (he's a serious screen addict).
I thought in the past I could keep him happy with like math circles and Vi Hart videos but he wants to learn new stuff--like, make forward progress rather than just play with the same ideas. Does anyone have any encouragement or warnings? What happens when an otherwise normal kid finishes high school math before getting to high school?
First off, I think there may be some diversity of opinion regarding what "otherwise normal" means. =)
Re your actual question: Many of us have afterschooled to meet needs not touched by the schools. You've already tried AOPS, which is a frequent favorite. We used the Singapore secondary curriculum (Discovering Math/Dimensions Math), of which the original non-CC version goes up through algebra II/trig, followed by Discovering Additional Math, which reaches some univariate calculus. You could also try playing around with MIT OpenCourseWare. There are OCW courses aimed at AP calc review for high schoolers, advanced/fun math topics for interested secondary students, and, of course, also full-on MIT math courses, the most comprehensive and organized of which are in the MIT OCW Scholar section of the site. While entry-level classes are the most thoroughly covered in OCW Scholar, pretty much any level of math that he might want prior to his college matriculation (whenever that may be) will be accessible in OCW somewhere. Also, check out the other resources for high school learners on the site.
And rest assured, there are others (admittedly, not a lot!) who have been through this. The path is unlikely to be quite typical, but that doesn't mean that the rest of his experience can't be balanced and age-appropriate to him.
...pronounced like the long vowel and first letter of the alphabet...
Using the AoPS textbooks is very different than trying out problems via Alcumus. Assuming you only did that, you still may want to give the books a try. The books are inquiry based but formal. You usually start with some problems and then develop the structure after trying to figure them out. (Ditto for the online classes)
Also, is he able to actually solve the AoPS challenge problems? I'd want to know that before writing them off as not interesting.
And then the topics he like are not particularly standard k-12 curriculum. (Many do show up in contest math btw.) So honestly a good math circle would be very likely to cover some of them and would not just be playing with same ideas. Have you looked at the Julia Robinson Festival problem sets?
In the end, there is no one right way to learn. So trying some of the alternatives seems like a solid plan.
Thanks for the suggestions! I will poke around at the things you mentioned.
We've done Julia Robinson problems, and he really likes them. I'm sure he'll continue to get joy out of those things. I should do some work to find problems that are matched with whatever he's learning (when I figure out what to teach him next).
He did a couple of math circles, but he talked too much and had to be shushed, and it didn't really satisfy him. It would be nice if he could be in one with older kids, I think. (Or more-similar kids, but there's no way we could afford anything like epsilon camp! Although that looks super cool.)
He is able to solve some of the more challenging AoPS problems, but he's not especially good at them or anything. It would probably be worthwhile to at least convince him to learn how to approach these kinds of problems though. I don't really know why he doesn't especially like them and finds them difficult when he doesn't find certain other things difficult (he will happily explain Monty Hall to you). Maybe because they look too much like story problems? Probably if I just called them "puzzles", he would be more into it.
We do have a couple of the AoPS textbooks (prealgebra, and algebra we just started), but we've just done problems from them (for review and looking for gaps) because he already knew the lessons. He prefers to learn from books like Life of Fred (when he was little) and Cartoon Guide to Whatever. And "popular math" books (like Things to Make and Do in the 4th Dimension). Basically he is motivated by things that teach while showing the beauty and awesomeness of math, things that are entertaining (pictures, jokes), or by feelings of achievement and progress (like khan academy--he didn't do much of that because he uses up his "educational" screen time allotment on writing music and programming). He also read things like Complete Idiot's Guides. Maybe I should give an AoPS book about something he doesn't know yet a shot. (They're so expensive!)
Anyway, thanks, lots to investigate!
One other thing I would like to teach him is how to do formal proofs of math things? Not geometry proofs, but more general like number theory proofs, I guess? I'm not sure exactly what I'm looking for, but this kind of thing must exist (you can tell I don't have much math education myself!). He has good instincts about proving things (he had an interesting idea for goldbach, for instance, which of course didn't begin to work, and he has proved various cool things by drawing diagrams) but I think he would enjoy something formal and I haven't found anything that lays it out for you.
So those with older kids, what did your kids do when they were high school aged? Did your kid find a mentor at some point?
I just have some random thoughts. First, I have no experience with the lower level AoPS courses or books so can't comment on your DS' problems with AoPS Pre-Algebra. However, I encourage you to try again later, especially with the actual classes although the books are great too, when your DS is ready for Algebra 2, Geometry, Algebra 3 and Pre-Calculus. While there are quite a few problems from competitions, those are not the only problems in the books or courses, at least the ones that I have seen. More importantly, AoPS courses force the student to actually understand rather than memorize at a superficial level because the students are deriving formulas and writing proofs. Also, once you are confident that your DS has mastered Algebra I, you may want to try the AoPS Number Theory and Counting/Probabilty courses/books.
It also may be that you started your DS too low with Pre-Algebra if he is already at a Algebra 2 level. As you are concerned with holes, you may want to use the short AoPS tests to verify that your DS has mastered Algebra 1 and Geometry. That way, you can then go with confidence to another curriculum for Algebra 2 and beyond.
I am not trying to sell AoPS as there are a couple of respected options already mentioned by other posters. However, your DS will eventually need some kind of high level math curriculum to supplement his school math curriculum if he is serious about pursuing math. There is nothing wrong with skimming concepts off library books if he enjoys reading them. In fact, when DS was his age and a bit older, I used to pick up random adult math books from the library and just left them around for DS to read or not as he wished. From your post, I am assuming that your DS is already reading adult math books from the library so perhaps continue in a low key fashion.
Since your DS obviously loves math, I am not sure you need to worry about "interesting and fun" since he likely already finds math interesting and fun. Perhaps look for a highly recommended textbook on Amazon and go through that quickly with your DS to cover holes?
You could try the murderous maths books. They are zany English books for about his age. The emphasis is on entertainment with maths thrown in. I (as an adult) learnt a few things from the them so there should be something in it for him.
The English culture of maths is different enough from the American that there will be stuff he hasn't come across.
Also, the (English) Numberphile youtube channel has some interesting stuff on it - definitely not aligned to school but just interesting e.g. the enigma machine
Seems like your kid enjoys more logic and less formal approaches.
Have you thought about a cheap computer and getting him programming?
It is the ultimate sandbox to play in.
The Raspberry Pi is cheap and the reaspbian comes with Python already installed. Other languages and IDEs can be installed fairly trivially. Of course, you would need a monitor and keyboard too but they are pretty cheap too.
Thanks! I guess the first thing I'll do is see if I can find a good level of AoPS to start at and try it. We haven't done much lately. I think I need to ask his math teacher if he can just not do his math homework (lately, pages of dividing decimals--useful only in that he gets careless when he's bored) so he has time/bandwidth to do more appropriate stuff.
He does like programming, although he doesn't know too much yet. He has written cool programs in Javascript for mathy things like randomly selecting 20 numbers between 1 and 40 (which turned out to be much more interesting and difficult than it sounds, at least for a beginner).
Raspberry pi might be a good holiday present. (although what he really wants is an $800 synthesizer. Too bad, kid. Get a job.)
How old is your son now, Portia? What was his first proof? What does he work on with his mentor? My son's first proof was (x+y)(x-y)=x^2 - y^2. He didn't know how to multiply out (he was 6, or maybe 7). He drew some rectangles. (First he noticed that (x+1)(x-1)=x^2 - 1, but he never tried to prove that one.) My oldest son also came up with proofs when he was little (but older, maybe 7 or 8), but he always did it with words. Little guy is very diagrammy. When he was a toddler, he spent most of his time when out in public talking about floor tiles (and yet he is the most socially skilled of all my children (which isn't saying much)).
