Proof-based math classes came like a revelation to me. When I took Real Analysis, for the first time in over a decade, math was fun. You weren’t just memorizing and reapplying recipes. You were seriously thinking about unique problems and devising solutions. And all the while, you were learning where all these techniques actually came from and how everything connected together.
I don’t understand why we can’t have more proof heavy math in high school. Who cares whether you remember the arctan substitution or whatever in an integral; I’d always just use a solver for that anyway. I’d rather be learning about what an integral is in the first place.
After teaching proof-writing to my students for several years now, I've seen a lot of variation in how quickly students take to the skill. Some of them have the same experience that it sounds like you and I had, where it "clicks" right away, some of them struggle for a while to figure out what the whole enterprise is even about, and everything in between. Basically everyone gets better at it over time, but for some that can mean spending a decent amount of time feeling kind of lost and frustrated.
And this is a very self-selected group of students: they're all grown-ups who decided to spend their money and spare time learning this stuff in addition to their jobs! For the kind of high school student who just doesn't really think of themselves as a "math person", who isn't already intrinsically motivated by the joy of discovering what makes integrals tick, I think it would be an even harder sell. High school math teachers have a hard job: they have to try to reach students at a pretty wide range of interest and ability levels, and sadly that often leads to a sort of lowest-common-denominator curriculum that doesn't involve a lot of risk-taking.
proof based math requires critical thinking and its a lot harder to scale the teaching of critical thinking. We dont' pay enough for teachers of quality to be able to do this at the public school level. Its also much harder to test for in standardized tests.
You could test it using interactive proof verifiers. This would also make it a lot easier to teach, since proof verifiers can handle even very complex mathematical proof via the repeated application of a mere handful of rules. (The rules are also surprisingly similar to the familiar "plug and chug" workflow of school-level math, only with different underlying objects - lemmas and theorems as opposed to variables and expressions.)
£30 for 6 months is pretty damn cheap and you get to keep it forever!
ebay example of the latest edition for sale: https://www.ebay.co.uk/itm/197011707080
On archive.org too if you are happy with PDFs: https://archive.org/search?query=creator%3A%22The+MU123+Cour...
First MU123 book A: https://archive.org/details/BookAMU1232ndedOU2014/MU123-Book...
This is a proper accredited course developed over 50 years or so with its own textbooks and material from a respectable university, not a gamified subscription portal experiment put together by god knows who that can disappear in a puff of smoke at no notice.
I can second this recommendation. The maths books are _excellent_.
It's hard to explain how, but let me try: most of the maths textbooks I possess (plenty of them) are written with the assumption that you attend lectures at a classroom and use them for extra material/exercises/reference.
The OU books are written with the assumption that you learn from them as the primary material, so they go a lot further with regard to explaining things as well as producing them from first principles.
- 5th grade math
- prealgebra
The book does look high quality. But I'm surprised it covers these fundamentals, given it's for a university course.
If you are above this level, you would start with the "intensive start", which skips MU123 (allowing you to pick another module in its place) and then starts with MST124 (precalculus, trigonometry and single-var calculus, roughly), moving you on to MST125 (intro to proofs, number theory, more calculus, linear algebra, etc), in a faster pace.
But I guess there are people who will have forgotten that stuff, if they're not using much math day to day.
If you can and willing dedicate on average 2 hours a day (a big commitment but I think I was able to hold it for several month with them) the cost of mastering, say, Linear Algebra will be ~4 less then if you subscribe and will be spending ~30 minutes a day.
I guess it depends on where you are at in the world, but in our neck of the woods $50/month is an absolute bargain compared to using a tutor. Not to mention you get to work at your own pace and to practice spaced repetition consistently.
> I can't help but feel that the people who would benefit from it the most are also the people least likely to be able to afford it.
Even if it were $10/mo, the people who would benefit from it the most (around the world) still can't afford it.
$49/month is almost nothing to me now, but it would be prohibitively expensive for a 15 y.o. me in freshly independent Czechia.
I suspect it would also be prohibitively expensive for most 15 y.o.s in the developing world today, and these are the guys and gals who stand to gain the most.
I wish there was PPP for the subscription, i tried for a few months but stopped the subscription recently.
the format basicly consists of a list of modules (lessons or reviews) to work though. each lesson module starts with a short test on how to solve the problem it teaches, 1-2 worked problems and 2-3 problesm then more short text on how to solve some variation of the problem coupled with 2 solved problems and 2-3 for you to solve. review modules just give you a few problems to solve. if you can solve 3 without making any mistakes, it will finish the module and you can go tot he next one. every few days you are presented with a quiz that tests what you know and genearates review modules for you to go though before retesting you.
overall, it seems spartan but its effective. you spend most of your time actuvly engaing with the problems with just enough information to solve them. the structure of the courses are setup such that you learn and master prerequisites before you are presented a lesson so there isn't much of the frustration you might find where a problem implicitly assumes you know something you don't. That a big reason they start you with a placement test.
overall I'd say the $50 price tag is worth it. its very efficient vs reading and working though problem sets where you don't necessarily know what you need to know to solve them. you're not spending time figuring out what your knowledge gaps are as it fulls them in as you go for you.
https://www.mathacademy.com/terms-of-service#cancellation
It's what I did 10 days ago before deciding to try it out
I personally just put a reminder in my calendar for all such things and be done with it.
It took myself 2 1/2 months to complete Mathematics for Machine Learning on Math Academy last year (2024) working through reading material, taking notes, and completing all the exercises took all day everyday I loved it, this was after I completed Khan Academy (starting from the beginning of mathematics negative numbers, to the end differential equations) because I kept putting it off for years when I got to busy.
The main thing for me was learning not to get too frustrated when getting an answer wrong. If I made a mistake, I focused on understanding what went wrong, looking up youtube videos on the topic if it was confusing, and then trying again.
At the end of a lesson I wish I had someone to bounce questions off of but thats when I used chatGPT.
Congrats!
Good Luck with M4ML its a great course! Covers a lot, I was impressed, wish there was some videos or more visuals but it doesn't hurt to use youtube. I took maybe 7 pages of notes on my github and each over 4000-8000 lines (I used the notes to do the step by step exercises it was easier for me to type notes and do the exercises on computer than pen and paper this is what I used to do).
I take the notes because I will probably forget, but I think its key to always be learning and keep practicing even when your done the course.
Once I get hired again I will def take Discrete Mathematics. In the mean time I've just read books on ML and LLMs, free online courses, youtube videos etc.
My middleschool principal thought it'd be a good idea to skip me over pre-algebra into alg 1.
Turns out that doesn't work great, and I still have confidence issues because I have a hard time remembering the properties of addition & multiplication by name. I know the rules.
My middleschool principal thought it'd be a good idea to skip me over pre-algebra into alg 1.
1. Read each sentence at half your normal reading pace
2. Skip every other chapter.
Sounds ridiculous, right?
That's my reaction when people propose grade skipping as the only solution for a child whose natural pace is 2x the 'standard' pace at which math is taught in school.
1) Allow the child to go at their natural pace.
2) Grade skip every 2 years, with take-home exercises.
3) Grade skip every 2 years, without take-home exercises.
4) Force the child to go at the same pace as the rest of their same-age peers.
That was my biggest problem, that and I wasn't actually on-board for the skip. Educators need to learn how to admit when they fucked up and learn how to improvise a new strategy.
skipping chapters of a novel doesn't work very well, but it works great for the encyclopedia, and pretty well for a lot of textbooks
it's also not that hard to use khan academy or wikipedia to fill in the gaps, if you did miss something
I really like the way my kid's middle school does it: accelerated 6th grade math covers the entirety of the 6th, 7th, and 8th grade standard math curriculum, which sets the kids up for algebra in 7th grade and geometry in 8th. Because the standard middle school math curriculum is essentially just advanced arithmetic, it's pretty straightforward to bundle this way. It also makes it easy to inject 7th graders who missed 6th grade accelerated math into the accelerated track if they pass the algebra qualifying test before 7th grade.
When I was growing up the G&T program started in 4th grade and cohorts from multiple schools were pulled into a school that ran the "gifted" program. Essentially all the kids were tracked from 4th grade through high school graduation and there was no real possibility for non-G&T kids to get into the "gifted" classes in middle school. In HS that just transitioned into APs and college dual-enrollment; by the time I graduated HS in '99, I had 22 credit hours of college classes banked, including dual-enrollment bio and calc 1 + 2, plus a bunch of humanities APs.
Today -- at least in our bay area public high school -- there's no tracking outside of math and the vast majority of classes can contain students in multiple grades. That absolutely was not the case when I was in school, and imho it's an improvement.
Filling in gaps is fine for people with good study skills, but that excludes the vast majority of elementary school students.
whatever the case is, i think the idea behind skipping grades is that the kid isn't learning much in the classes they're in. they may not learn much in the next level either, but it allows the school to test that they've learned what they were supposed to (from class or elsewhere), while wasting less of the student and teacher's time
that said, testing out seems like it'd be better than forcing the kids to sit through yet another math class, even if it's one level higher. more time to touch grass, or read in the library, etc.
Over the past few years, while homeschooling my daughters, I've come to see the way math is usually taught as horribly pathological. In the US, where we live now, it's often seen as a competitive activity -- almost like a sport. In the UK, where I grew up, that wasn't the case but still it was taught as this huge body of knowledge and skills with almost no motivation.
My daughters are so advanced in math and I really don't believe it's even mostly due to innate ability. It's because, just to take an easy random example, when we studied geometry our very first lesson was me pointing out that the word "geometry" just means "earth measuring", and it was useful for farmers to be able to do that. Or, when we proved the irrationally of sqrt(2), of course I entertained them with the tale of Hippasus being thrown into the sea by the Pythagoreans. For basically everything we've learned there are so many fun stories. It makes me sad that most students of math never get to hear them.
Small companies have to understand the value of local pricing — nobody is willing to pay above h percent of their salary for a service X, and there's only so much that rule can be bent. I understand that, at the end of the day, the company still has all their expenses in USA prices, but for digital services with no manufacturing or logistic costs, it can be better to make a modest profit than none at all.
It would be impossible for me to have one-on-one tutoring for a year at only €465 ($499 but I'm in EU). And that's regardless of the tutoring quality
I haven’t really looked at math academy, but I was in school (including college) I probably learned 40% of math from khan academy, 40% from textbooks, and maybe 10% from lectures.
How does math academy compare to Khan academy?
I didn’t, at the time, appreciate how challenging a problem it was until I started researching Bayesian Knowledge Tracing. While their definition of a skill can be a bit narrow, thus putting more time into reviewing things I'd rather move on from, it does work from what I've observed.
I recall they had a course on Abstract Algebra and other more advanced subjects, so if you're really interested, the great thing about subscriptions is that you can afford to try it.
Another good book for the author and others is "5 Elements of Effective Thinking" by Burger & Starbird. It thinks about thinking which can sometimes side step the depression of suddenly not thinking you know anything about anything that accompanies that big drop off mount stupid.
Depending on how senior they are at work, that can be quite traumatic. A lot of people in tech sort of base their self image on how smart they perceive themselves to be with respect to their peers. When that perception inverts their own world model makes them feel worthless.
In the two cases where people I was managing this occurred (that I knew of) their productivity dropped like a rock and they became seriously depressed. One I managed to get back on track, the other left tech and I have lost track of where they ended up.
Second - I love the website. It reminds me of what I think of as the golden age of web design where sites were mostly server side rendered with a little jquery / Ajax sprinkled in, and more information density was preferred.
My only desire is that their site worked on my phone- it would be nice to do a lesson when I have some free time and some paper.
Re: Math Academy, I used the service for ~3 weeks last year from a post here on HN by the guy responsible for the AI/ML knowledge graph behind the platform (I believe his first name is Justin). I was "only" doing about 30-60 minutes a day (a little bit higher than their guidance, but low for someone not doing math otherwise IMO).
N.B. Due to substandard early instruction combined with being "gifted and talented", I was placed by the test into Math Foundations 1 (or 2?). For example, I still don't have an active/working mastery of the unit circle. So if you're a real whiz, YMMV.
I found Math Academy effective at showing me my weaknesses and sharpening those skills in the short term, but I probably didn't do it for long enough to benefit from the spaced repetition effects. I found the UI/UX better than Khan Academy (sans AI), and much less tedious (when I demonstrated understanding, the questions moved on or increased the complexity vs. doing the full problem set no matter what).
When I cancelled within the first month to receive my refund (see other commenters mentioning the high price), I was surprised to see my support email and refund request email both went to one of the founders (or owner?), Sandy Roberts, who was emailing me while also attending her daughter's college orientation (or helping her move, can't recall right now).
Cancelling was painless once I realized I was getting a response from someone at the platform --- so if you're interested in trying it, I can recommend giving it a shot. Maybe there's some sort of economy for them if more (adult) people sign up, because 50 USD still feels a bit steep.
I understand that everybody has different financial circumstances, but personally I find it so odd how people prioritize their spending. $50/mo to level up your math game? Too much. 8x $6 lattes per month - totally worth it. $200k+ for a university education after which you STILL won't know basic math (or much else useful for most majors) - super totally worth it.
For me I'm just willing to pay a lot more than other folks are to learn interesting skills. Math, sailing, music, leatherworking, perfume making, whatever - to me that's such a good use of money.
The near-term costs vs. the long-term payoffs of learning the math skills I was learning were pretty clear (immediate costs, opaque long-term benefits other than being better at math in a general sense). I didn't have anywhere to apply the math, so I decided to spend my time learning more from sources like "Structure and Interpretation of Computer Programs", which is zero-cost to me.
I don't drink any $6 drinks of any sort; I also can't go back to change how much I did or didn't pay for a university education that did or didn't involve learning math for my major.
Other commenters I read who shared price concerns also didn't mention their latte habits or cost of their advanced degrees, so I didn't find this comment very helpful for making that comparison in my specific case.
The SQLite course was in a very different video format and took roughly 20 hours, but I learned a lot and immediately used that knowledge in two software projects that would've seemed insurmountable to me.
As for MA, it's taken a lot longer and has been difficult. I'm now at 6000XP and halfway through Fundamentals II. I have a lot of thoughts, but I (kind of) plan to (probably not) write a review after having completed Foundations I-III, since I haven't talked to anyone else who's done so so far.
https://highperformancesqlite.com/
The course content could all be learned from the excellent SQLite docs, and in greater detail. However, I think paying the $200 was worth it for me, since the course led me through a structured learning path, this is my "year of structured learning", after all. I had some minor complaints, but in the end I learned a ton and feel empowered because of it. I'd like to take his Postgres course too but the price is too high for me.
As an aside, all Dunning and Krueger showed is that everybody thinks they're in the top 1/3 to 1/4. (At least everybody in undergrad school at Cornell.)
The DK experiment depends on people ranking themselves against the cohort, other people who they don’t know. The DK effect probably doesn’t exist, it has been argued compellingly that the paper does not demonstrate what it claims to.
“To establish the Dunning-Kruger effect is an artifact of research design, not human thinking, my colleagues and I showed it can be produced using randomly generated data.”(!!!) https://www.scientificamerican.com/article/the-dunning-kruge...
“the asymmetry reported by Kruger and Dunning actually goes away, and even reverses, when the ability tests given to participants are very difficult.” https://talyarkoni.org/blog/2010/07/07/what-the-dunning-krug...
“The Dunning Kruger Effect is probably not real” https://www.mcgill.ca/oss/article/critical-thinking/dunning-...
Every time I read the DK paper, even now, I’m struck by how wild it is that so many people refer to this as a human-wide cognitive bias, when there was so little evidence to begin with, the study was so small and biased, so many people have called out problematic issues with their data and experiment design. If randomly generated data shows the same effect, then what, exactly, did D&K demonstrate?? https://digitalcommons.usf.edu/numeracy/vol9/iss1/art4/
Reeks of a publicity stunt.
But I gave up during the diagnostic test. It was very, very long, and didn't seem to be adjusting in difficulty, and asked similar questions. I'm normally a fast test-taker, but after about a third, I figured it would take me an hour and a half or two hours more.
I hope they've updated it by now.
To some extent, though, I don't think you're meant to spend lots of time on each question struggling to figure it out as it's trying to determine what bits of the course you can skip, so if you can't answer a question relatively easily it's best to just say "Don't know" and move on so you get some revision during the course.
Are there any other recommended websites for learning math (apart from Khan Academy, Math Academy)?
Math Academy looks very enticing!
Compare and contrast with actual one: https://en.wikipedia.org/wiki/Dunning%E2%80%93Kruger_effect
Sure, novices overestimate their skills - but the relationship is monotonous, with no peaks or valleys at all.
From Parent: Expensive Alpha quality - missing absolute value signs, auto-marks "1./2" as wrong compared to "1/2", parental controls have bugs, eg you want to pause, then change the date won't update the date you're pausing to. Uses XP (experience points) as gamified motivator and then doesn't respect their value by docking unnecessarily or due to bugs. Emailed on sign up to be "personally welcomed and invited to respond with feedback, or any questions" So I did, because I want this kind of effort to succeed. Response totally ignored all content of the email and suggested he could delete the account for me(!)
From student, who is quite a way ahead of his peers in math. "boring, annoying and stupid"
Account is paused due to the above, if we cancel it I'll be pretty annoyed given the unused portion and expense.
Khan Academy has very similar merit. You can donate a lot less than USD$50 / mth.
Maybe we got unlucky and the good intentions of "personally welcoming and requesting feedback, inviting questions" was lost due to getting snowed under so response LLM happened and totally ignoring follow up rather than being deliberately rude? Could well be.
Lotta hype about it, justified? I didn't see it.
It's a mere 15.5 pages of actual text.
https://en.m.wikipedia.org/wiki/Dunning%E2%80%93Kruger_effec...
I do see other graphs that tell a different story. Namely, that confidence is a monotonically increasing function of competence. If the data supports the idea that there is a valley of despair where confidence decreases as competence increases, I must be missing it.
https://commons.wikimedia.org/wiki/File:Dunning%E2%80%93Krug...
[edit: yes, it isn't currently on the Wiki page. On the other hand, I've seen that graph associated with that work before]
Dunning-Kruger described a relationship between people's subjective opinion of their skill, and their performance on a test. They find the subjective curve is less steep than the objective one (low performers believe they are closer to the center than they really are, and so do top performers). There's no "peak of stupid", or anything else on that graph.
Repeating vague associations you've seen on the Internet before is how misinformation spreads.
Either my eyes skipped past it or that dispute notice was added after I linked the image. Regardless it belongs there.
I have previously seen a similarly shaped graph with Dunning-Kruger effect discussions many times, including on Wikipedia I believe. Now I'm curious what the source of the misrepresentation is since it does not appear quite derivable without artistic interpretation from the paper's data.
Regardless, I'm glad to update and add to my beliefs.
Please note that despite the implication that seems to be in your final statement, I did not mean to say the graph was correct, only that it is a graph commonly associated with the paper's message and thus understandable for the author to have used. From that, the use of it doesn't quite come from nowhere. I'm fact, I didn't really say much at all. While Wikipedia is the first search result, the Decision Lab is next which has a similar, even more distorted graph on their page [0] and yet is a fairly well esteemed organization.
Glad to improve my knowledge but that the graph is in common use is not misinformation even if the graph itself misinforms and isn't from the paper.
Maybe the author is running some kind of A/B test between the actual Dunning-Kruger paper graph and the fake one?
Parents can make this worse but it’s pretty hard to prevent it.
Moving from Slovenia to SFBA in my mid 20's (~2015) was ... super fun like that. Sooo many people here are that most brilliant super talented engineer/founder/whatever from their home locale. But here we are just the norm.
Then you move or have some experience that opens your eyes and you see that there are so many people out there that you're not actually as special / smart / talented / athletic etc as you thought.
I've had the experience a few times myself, and it's always a bit of an existential wakeup call.
Related - About 10 seconds into my first job I decided that staying quiet for a bit and listening to people around me would be a very good strategy throughout life.
Very similar story here! Grew up in the Netherlands, joined an SV company in my late twenties. Huge mix of imposter syndrome and sadness that I hadn't been able to experience such brilliant people earlier.
Very much regret not choosing different educational paths that could've let me surrounded by them a decade earlier. I would've enjoyed life much more. On the other hand, I don't think either of us could've realized it before experiencing it first-hand, so no reason to beat oneself up over it.
Get students to build their own drone with MCUs - so they're forced to overcome challenges and practical trade-offs.
Introduce algebra in primary school, and calculus much earlier - so Green's functions, etc. can be taught in high school to those who want to study mathematics, and they feel what it's like to struggle to master concepts early on.
Hopefully with more of a shift to online courses and AI, this will be possible. Unfortunately the majority of schools just act more like a daycare centre / prison.
algebra and calculus is college level century ago, we already bring it on HS level in some part of the world (Asian country) we already learn it earlier than most of the world but that's not sustainable
And then after a few years you learn that you can accept the workload but it's not enough to guarantee everything will be OK. In school everything is handed to us, really.
My son (7 years old) is gifted in Math and as a parent I find it extremely hard to decide how much I should push him (register him to math competition, weekend math club ...) and how much I should just let him get 100% on exam and not accelerate the learning.
The best way to guarantee a gifted kid wastes a lot of their potential is to be in an environment that is too easy. It creates a devastating mental habit that won’t trigger until later in life, like college. Whenever they try to do something that doesn’t come easy, their brain will try to shut down out of a kind of frustration. They won’t know how to overpower it. It will cause depression, anxiety, shame and low self worth later on. Because the gifted kid will know they are wasting their potential, but blame themself for not being good enough to deal with it. It feels like being broken.
All of this is created by being rewarded for maxing out the rewards of a trivial environment. Someone needs to patiently and compassionately teach them to value overcoming appropriately sized challenges. To find and operate on the edge of their potential and ask for help to operate beyond those limits.
So yeah, grit and asking for help. Intelligence is mostly wasted without it.
- your child has a wall. At 7 he is not hitting that wall.
- that wall is probably mostly related to the pure math concepts, and probably less to his actual age when he encounters them. This is my assertion and I cannot prove it but let’s assume it is true. Precalc or calc is a typical wall moment, but for others it might be geometry or trig.
- one response to an eager math learner is to move them through the curriculum faster. They are happy, because everything is fun prior to the wall! You get to be the parent of that kid who is great at math! Let’s put the pedal to metal!
- what acceleration means is that your kid will hit the wall at 13 instead of 15, or 14 instead of 16, etc.
- those two years can make a big difference. Accelerating might be positive, in that they hit that at an age where you can support them better. It might be negative, in that they now have a crisis that their peers can’t relate to. Not accelerating might mean that they respond to the wall by pouring their energies into age-appropriate activities instead, like listening to loud music or being grumpy.
So no easy answers here. We did not think ahead clearly, and pushed forward, and had some decisions to make later. In retrospect I think it turned out fine, but I wish I had known that I was pulling the wall forward in time.
Moving through the curriculum faster is a common approach but it's also risky, because that's how the gaps are created that can then hinder your understanding later. Of course if you have reached true mastery of a given topic, moving forward is preferable to being bored to death, but assessing whether that applies can also be difficult at times.
My own kid went to MathPath (middle school camp by same people as Epsilon Camp). Loved it. “Yes, dad really, I want to spent a whole month of my summer doing math.” The social experience is great for kids to be with other kids that like math.
I'm a thoroughly useless adult, so it was a waste of money on their part, but it does happen. Or at least it used to.
No one told me that math is really 90% about writing proofs, all those homework problems I did were just the weed-out stuff, the academic equivalent of Leetcode.
So when I got put into some “real” academic math as a teen, I crashed and burned hard. I didn’t have a tutor and it never would have occurred to me to ask for one, so that was that.
When I was 18 years old in my first year of college, after my first semester grades came in, a guidance counselor set up a 1-on-1 with me to talk about the Rhodes Scholarship process and what my research interests were.
My response was: 1) what the heck is a Rhodes Scholarship and 2) how could I possibly have “research interests” as an 18 year old college freshman.
That was the final chapter of society considering me “gifted”, but it was just as well, I couldn’t imagine any greater success beyond getting a job and being able to afford my own apartment.
Mostly because a lot of my personal interests/ability to self-develop was related to Internet access. (My parents made VERY QUESTIONABLE financial choices and opted to pay for Internet access instead of food or clothing so I might have been freezing and my clothes all had holes in them but I could go online to talk to other smart kids.)
Also because I remember me + my parents being sat down when I was in elementary school and having my options talked about. In middle school once I was proven to have programming and math aptitude during the dot com boom, educational experts came to us and discussed specific gifted learning options (including things like private schools, skipping grades, or even pulling me out of school altogether for private instruction). None of this was initiated by my parents - it was brought to us. This was in the 90s.
I did teach myself programming in the 90s, after my friend loaned me his floppy disk with all his QBASIC stuff. Then dabbled in PHP, MySQL, etc.
We had one computer programming class in high school and I never got to take it because I had too many other electives. I don’t think it would have done much for me by the time I could have taken it.
It never really occurred to me as a teen that I could use the internet for getting really good at academics or broader “self-development” - I guess I just cared about video games and making money. Parents’ attitude was as long as I was getting As and going to college they didn’t need to do anything.
Of course you still hit the wall later. But I see all the reports of how terrible it is to be gifted and am so grateful that my experience was different.
If I'm honest, I never ran into an intellectual wall. I did choose a comparatively 'easier' path, but that was more because I had a wide breadth of interests and choosing something easier meant I'd have more time to indulge my various interests. I was still getting interviews for tenure track positions out of grad school and when I did try to work post-graduate school, my first position was at an Ivy where I was the only one on staff who didn't come from an Ivy League school. (I was too lazy/too absorbed in my own things to do what was required to go to one.)
I ended up disabled in my last semester of graduate school - the 'wall' in my case is my body being unable to accommodate the social/networking demands of an academic or high powered private research career rather than my running into a topic I felt was beyond me. Particularly combined with being on my own in a HCOL area as that lifestyle required: Doing all your life management on your own with no safety net along with running at that high of an intellectual level is near impossible when you have a severe disability. (I have MS.)
I've been 'stuck' intellectually once in my life, and it was the result of a medication we tried for symptom management, and I found the feeling horrifying, if I'm honest. It was the first time I'd run into a problem where I had to sit there and think and still couldn't come up with a way to proceed, versus running into a problem and just being too damn lazy to bother. (Being able to see what I would do to solve the problem is very different from being motiviated to do so.) Apparently, most people feel that way fairly often? It made me way more sympathetic to people who didn't like school or who don't like learning.
If he likes to do math you make it available, if he would rather play with legos instead of doing math you let him do that in his free time.
You can encourage learning and problem solving without it having to be math, or pushing.
As you can imagine, there is a whole world of kids like your kid who love math and want to do nothing more than math.
If you're interested I can chat with you or recommend resources here if you decide to help your kid do more math.
All of this to say I have been writing software professionally since 2006, and while I do struggle with the thinking behind functional programming and math-heavy subjects like graphics programming, I have written lots of business software that has brought me personal satisfaction. I would really like to understand calculus better, but I'm not sure if it would actually do anything for my skills in programming. If math is holding you back, think about whether you need the full breadth of CS knowledge, or if you just enjoy writing software.
I became better at code organization, making code maintainable and simple enough to understand unless performance was an issue, and general people skills. I can understand why math and software are so close to each other, but at the same time, I don't think it needs to hold you back unless you really want to go into a topic that is deeply intertwined with math. It took me four times to get past pre-Calculus, and once I did, I realized that I just did not enjoy that type of math and didn't need it to build useful software (as in makes people's lives easier and/or generates profit for business) that I also find fun to create.
It genuinely wasn't until I was in my mid twenties that I wanted to look at anything mathematical again :)
