Not structured, but my kid at that age loved playing "magic function." Player A thinks of a function (starting simple, like x+3, and as the kid gets better at the game, polynomials like x^3/3+x-4, square roots, 3 dimensional functions, etc...). Then Player B gives values of X, Player A spits out what the corresponding Y value would be, until Player B guesses the function. Then switch roles. The guessing has a strategy too, since the kid will quickly figure out that you can get useful info guessing fractions, zero, negative numbers, very big numbers, etc... Playing that game orally while on hikes and walks was probably 90% of the early math my kid did.

Alaska uses a 'carrot' approach. Anyone can homeschool with zero oversight, but almost no one does, because if you sign up with one of the many official school district programs you get around $2000/yr per kid for educational expenses (more for high school), can check out a laptop or tablet, have the ability to take a class or two in the local school, participate in sports, optional homeschool field trips and contests, etc... In return, you have contact with an oversight teacher and have to turn in semester reports (pretty low paperwork requirement), and (pre-covid), do the same tests the public school kids do. I've never asked for much of anything other than the money, but the oversight teachers seem pretty friendly and helpful. Around 9-10% of Alaska kids are homeschooled in this system.

Thanks @bookbard. Things are great here. That little girl who used to love frozen just decided to go to public school for the last half of 4th grade, and my just-finished-6th grader is my only homeschooler. He's almost certainly going to homeschool all the way through. How about you? As for my mathy kid... I don't think I need to "cultivate" anything particularly - he'll like what he likes. I'm mostly just feeling a little angst that I don't want our scattershot rabbit-hole child-led approach to education (that's worked well for him so far), to cause problems for him as he moves towards the age of transcripts and grades and college. Most threads I read on here, even looking at advanced kids, talk about way more linear paths than he's ever followed. Kids that "did algebra" at some point, and followed some sort of designed progression of topics. My kid has either never done algebra, or has been doing it since kindergarten (when most of his math was playing the "magic function" game with polynomials while hiking). I feel like it might be easier if he gets himself to some distinct 'level' in a normal math sequence, to be able to take classes from other folks more easily, but I don't want to squash what's worked for him so far.

My kid has been working through an intro college physics textbook, which works well for him for the concepts and the problem sets, but wants more labs and experiments. Is there a good source of physics lab ideas, preferably broken down by concept? I'm happy to modify things and fill in the details, but it would be helpful to have even just a list of what sort of labs are usually done for each concept.

Thanks both for the suggestions. I sent him those contest questions and I'll see what he thinks when he gets to them (for some reason, all I found when I first looked was the 'AMC' contests -- I know nothing about that world). And I'll see if I can find any online talks he might like (he does know about you tube channels like numberphile, and 3 blue one brown). We absolutely do not have any in-person options, with or without a pandemic. I do want to let him explore and see if he loves math most, of course. I just also want to have enough of a clue in steering him that whatever choice he eventually makes, he's well enough set up to take that path.

Thanks, I'll do that and aim to sign him up in the fall. Summer doesn't work for us schedule-wise for a regular class -- we do far too much backpacking -- but we do school year round. As for the higher-level math contests, do you have suggestions? I'd particularly like to find proof-based untimed stuff from past years, with solutions available. He's not interested in competing, just in the questions. Also not interested in any question that he could possibly solve in less than an hour, so not in the timed competitions. The only place I've found good untimed questions is on summer camp applications (math camp and math path). He likes those, but they don't publish solutions, so I can't really tell if he's got it right in the end.

He hasn't done any contests, but I've found problems from old higher-level contests online and given him those to puzzle over just for fun. The AoPS class might be a good call for next fall, to have someone look at those proofs for him. He likes that format -- he's really enjoyed their Python classes. No idea how to choose which one -- do you think I should just set him to the "are you ready" and "do you need this" tests for the whole lot? Not sure whether it's better to aim high so he's sure to think the problems are interesting, or to aim low to not overwhelm him with the work load. I'm not likely to find any other sort of tutor -- we live in a very tiny town, and I'm confident that none of the 3 teachers (total!) at the local public school know more math than my husband does.

Maybe similar to what you're describing? He always seems to know the algorithms. He's good at problem solving, but I'm sure he can get better. I think he really needs practice in writing up and explaining his proofs better, but it's hard to help him with that when I'm often not sure if I really understand his proof, or whether it's correct. I have tried the challenge problems approach before. The only math book he ever "completed" (I don't remember when), was Beast Academy 5, where I made him do all the starred problems (he didn't touch any of the others). After that I got him to do all the challenge problems for the first four chapters of AOPS pre algebra, but he got antsy after awhile, and we moved on. Then he was desperate to learn calculus, because it was the only kind of math he'd heard of he knew absolutely nothing about. Perhaps algebra and geometry challenge problems would hold his interest, though. He says he's not at all interested in math contests, but he does like the problem solving involved in the untimed type of contest puzzles, when I've found them online.

This isn't a simple question, but more a general request for advice -- my kid is 12 and I feel like I need to have my act together as he moves forward to make sure he's got a shot at doing the things he wants to do. Currently he says he wants to be a mathematician. He might easily change his mind, but I wouldn't be surprised if he means it. He really loves complicated puzzles and trying to come up with proofs, often for problems he's created entirely himself, and is happiest puzzling over the same proof for days/weeks/months at a time. And all the other things he loves (physics, computer programming), are very math-heavy as well. His formal math education has been super scattershot -- largely consisting of me giving him things that turn out to be too easy, then skipping ahead. This year (6th grade), he's been slowly working through the AOPS calculus book with his dad, with diversions into his various puzzles and proofs. It's the first math book he'll probably get all the way through, because he does find it quite difficult. He likes working on the edge of his abilities like that, but now I'm not sure what to do with him next? He's never officially studied algebra, geometry, etc... but has always seemed to either already know what he needs or be able to fill in any gaps by quickly looking up the relevant topic. I figure people here are most likely to have dealt with kids like this. How should I approach the next 6 years to set him up well? (both from a learning and a college admissions perspective), while not boring him with requirements?

Let's see. He'd rather do more interesting things with Python than learn another language. I'm not sure what going more into data structures or algorithms would mean (I know no programming!), but he does like understanding the concepts behind things. He likes math, science, and messing with data and graphs. On his own he programs things like little games, encryption/decryption algorithms, and ways to get the computer to do math with bigger numbers. He also has a raspberry pi with some electronics pieces, and has been doing some coding related to that, with projects like morse code lights, photo-sensitive motors, etc...

Any thoughts on where to go after the AOPS programming classes? My 6th grader is taking the intermediate python class now, and really enjoying it (though the workload is a lot for him -- these are his first online classes ever). He learned a little programming on his own before the classes, but the classes seem to have accelerated both his interest and skills, so it'd be nice to continue that momentum.

Related question, but what do folks think about using Alcumus without any book? My 10 year old never even opened his Beast Academy guides, and just did problems, mostly the starred ones. So since he finished that, I set him up with Alcumus pre algebra a few days ago, and he's happily worked through several sections. If a kid prefers to do math without being taught it in any way by a book or human, will he have problems with that later?

Crash Course on Youtube? They have lots of thorough video sets, aimed at high-school AP students I think, and lots of different subjects. Quick-paced, fun, good for a kid who remembers things easily. My kids were huge fans of their Anatomy and Physiology course.

Yes, my kid usually was asleep for the lessons too, but it was totally fine to watch them later. The tournaments happened later in the day, which were the only things he really needed to be live for.

OK, so I'm totally promoting something my kid loves, but I think there might be other geeky game-type kids on here who might love it too, so I will. The more kids, the more fun for all. Homeschool Chess Club They run everything through a secret FB group and through a club on chess.com A chess coach in Arizona started an online club for homeschool kids and has been running a "summer camp" that my son's been part of for the past couple months, with live online tournaments, video lessons, a FB group with discussion and puzzles, and lots of opportunity for kids to challenge the other kids. The kids range from beginner to pretty good and I think most are 6-13 yrs old, with a lot in the 8-10 yr old range. Video chess instruction and puzzles are pretty easy to find online -- kids to play games against are not so easy to find, so having the club has been great. There's going to be a fall semester too (we'll do it again for sure), and I think it's cheaper if you sign up in the next couple weeks.

Science writing is a good one. I was a PhD student who found primary research a bit too esoteric after awhile -- working on things almost no one would care about. I have found magazine-style science writing to be more fun and more influential. I love the novelty of being able to dig into a critical issue, read up, interview folks and then write something engaging that educates the public.

Crash Course Anatomy and Physiology (on YouTube) is really good. Quite in-depth. I think it's probably aimed at high schoolers? Each body system is separated out into several videos. My 8 yo loves it. It does include a the entire reproductive system, so my two kids are rather well-informed for their age.

I find that reading adult-targeted popular science aloud to my 8 year old and 6 year old is quite effective -- better than kid-targeted books, usually. National Geographic articles would be fine in audio format for them, though neither would be able to come remotely close to reading one.

My 8yo is a science whiz and a struggling reader (likely dyslexic to some extent). He would be reasonably likely to correctly read something like "neurotoxic venom" in a passage about snakes, since he's interested in that kind of thing and has learned it in other ways. That said, he'd also likely mix up things like "net" and "next" and run out of steam after a few sentences. I think long sentences are a much bigger barrier than technical words for a struggling/new reader on a scientific topic. It takes so long to get through the whole sentence that they're unlikely to be able to correctly untangle all the clauses.

I have a science nut (just turned 8) who can't read. Probably dyslexic, but I'm not totally sure. He has enjoyed chemistry the most, and has primarily learned through videos, websites and simulations, combined with a molecular model kit, experiments, and me reading popular science (adult) books out loud. There are a lot of good videos out there. Some favorite YouTube channels of his are "Periodic Table of Videos", "Crash Course", and "Amoeba Sisters", but he browses related videos a fair amount too. He's enjoyed the simulations on pHet, and has spent many hours on ptable.com. We have a huge collection of molecular models, and the Mel chemistry kits (though we just had amazon purchased chemicals and some beakers before -- that worked too). He's taught me science I don't know, so not reading hasn't been a big issue.

How about a real world example, where negative numbers are used to measure real things? For my kids, the tides (we live on the ocean). If the tide is dropping from +20 feet to -4 feet, then it's quite intuitive that the difference in water levels will be 24 feet, not 16. And difference is subtraction, so 20 - -4 = 24. Temperature could do the same. Say it was +10 degrees during the day then dropped to -5 overnight. How much did it cool down? I think these are less abstract than debt, and make the answer more obvious.

Has he tried the simulations at pHet? https://phet.colorado.edu/en/simulations/category/new Most are pure simulations, with a few set up as games. My son has enjoyed a ton of the chemistry ones, and a bunch of the physics ones as well. If I remember right, radiocarbon dating is set up as a game. They've got a huge array of them, mostly high-school level science but ranging elementary to university level, across all basic scientific fields plus math.

Good to think about what your kid needs, but I wouldn't put too much stock in the Prodigy grade levels. My 7yo is happily enjoying the 6th grade prodigy curriculum, and I think he's reasonably good at math, but no genius. When I look over his shoulder I see that the questions have the same format over and over again -- no puzzles, no tricks. So if you can figure out the area of a triangle once, you can keep doing that until the program decides you've mastered it.

OK, I apologize for my overly dismissive review. Once Prodigy was done giving my kid a million and two place value questions (who knew any 6th graders are still learning place value?), it did actually give him more real math. And whether or not the Ontario 6th grade questions would challenge real 6th graders, most of them seem to be a decent level for my 1st grader, so the program does adapt to him pretty well on its default settings. I only have to pop over for 1 min explanations now and then. And the cheesy format of wandering around fighting monsters and catching cute pets is a big hit for the 7yo age group. My kid also liked both dragonbox apps and hands-on-equations (think he got bored partly through level 2 because he thought the white pawns were going to be another variable, and was disappointed there was still only one variable). He doesn't like paper-and-pencil math nearly as much, though he liked Beast as a read-aloud.

Verbal math puzzles are fun. Our favorite is "magic function". Person A comes up with a function. Person B says a number. Person A says what the function would spit out. Repeat until person B can guess the function. Switch roles. My 6yo played this every day for awhile, and it can comfortably span the range from functions like "x+2" to functions like "the square root of (x-2)" Biggest plus is that kids can stump parents with these also. Another one my son really liked (we played a lot of these on a long hike when he was 6) were "bottle problems". Pretend that dad has a bottle twice as big as moms, which is twice as big as the kid's, which is twice as big as little sister's. Say little sister doesn't like her juice and pours it all into dad's bottle -- how full is it? Say mom drinks half her water, then splits the rest evenly between the kids. How full is each of theirs? Limitless possibilities, and you can make up funny stories with them. Later, we ended up talking about infinite series with this one, adding leaking bottles, etc...