You Are Not Dumb, You Just Lack the Prerequisites

Lovely article, though honestly getting those prerequisites also takes a lot of time, effort and either motivation or discipline in ample amounts.

As someone who was the “smart kid” growing up, going to the university without good work ethic was pretty eye opening, no longer being able to coast on intuitively getting subjects, but rather either having to put in a bunch of effort while feeling both humbled and dumb at times, or just having to sink academically.

Even after getting through that more or less successfully and having an okay career so far, I still definitely struggle with both physical health and mental health, both of which make the process of learning new things harder and slower than just drinking a caffeinated beverage of choice and grokking a subject over a long weekend. Sometimes it feels like trying to push a rock up a muddy slope.

And if I’m struggling, as someone who’s not burdened by having children to take care of or even not having the most demanding job or hours to make ends meet, I have no idea how others manage to have a curious mind and succeed the way they do.

Admittedly, some people just feel like they’re built different. Even if I didn’t have those things slowing me down as much (working on it), I’d still be nowhere near as cool as people who dive headfirst into low level programming, electrical engineering, write their own simulations, rendering or even whole game engines and such. Maybe I’m just exposed to what some brilliant people can do thanks to the Internet, but some just manage to do amazing things.

A really fascinating corollary I've observed since I've gotten into more advanced maths, or even doing actual research as a PhD student, is that there's nothing special going on at the higher levels. You're just working with different materials. Materials that require more time and effort to 'get', but once you get them they are just another tool at your disposal.

I was similar to the author in that, throughout high school and undergrad, I presumed that the mind that could comprehend advanced math or do novel research (in any field) was truly unknowable. Like there was this x-factor they had that wasn't there for me.

I've long enjoyed puzzle games (like The Witness or Stephen's Sausage Roll). It turns out that problem solving in non-trivial domains is never terribly different than problem solving in those games, or any other domain really. Like my brain isn't doing anything different than the usual tree-search algorithm that any chess player performs when they are projecting moves ahead into the future.

Its just iterating on concepts that seem abstruse to most people. But at the end of the day, deep problem solving in math or AI research tends to be the same moving-shapes-around-in-my head that I would do if I was trying to move an awkwardly shaped couch through a narrow doorway.

Just thought I'd add a comment as someone who came top of the state in my grade in multiple olympiad competitions:

I always felt that a large part of my advantage came from having a strong understanding of maths from the ground up.

I felt that a lot more people could have gained the same level of understanding as I did if they had been willing to work hard enough, but I also felt that almost no-one would, because it'd be an incredibly hard sell to convince someone to engage in years-long project where they'd go all the way back to kindergarten and rebuild their knowledge from the ground up.

In other words, excellence is often the accumulation of small advantages over time.

I'm a tutor, mainly working with adults who want to learn proof-based math, and the message behind this post definitely lines up well with my experience! If you're the sort of person who's animated by the idea of learning math but finding it challenging, it's worth considering that you might be missing some knowledge or skills that you'd be able to develop just fine if you knew to focus on them.

There definitely is such a thing as "mathematical talent", but (a) if you're really excited by math then there's a decent chance your limiting factor is knowledge rather than talent, and (b) there's plenty to appreciate in the subject regardless of how much of it you have. My students come to me at all different levels but if they have enough time and motivation to work on it they all learn a lot of math!

There are also plenty of people in the world who just aren't that into this stuff, but that's not really the population I'm talking about --- unless they have to learn it for some reason, it probably doesn't bother them that much that they don't know a lot of math! And I imagine a good chunk (though probably not all) of this group could probably find something to like in the subject if it was presented in an appealing way.

From Neal Stephenson's "Cryptonomicon", at https://archive.org/details/cryptonomicon0000step_b9v1/page/... :

> “Shut up about Leibniz for a moment, Rudy, because look here: You—Rudy—and I are on a train, as it were, sitting in the dining car, having a nice conversation, and that train is being pulled along at a terrific clip by certain locomotives named The Bertrand Russell and Riemann and Euler and others. And our friend Lawrence is running alongside the train, trying to keep up with us—it’s not that we’re smarter than he is, necessarily, but that he’s a farmer who didn’t get a ticket. And I, Rudy, am simply reaching out through the open window here, trying to pull him onto the fucking train with us so that the three of us can have a nice little chat about mathematics without having to listen to him panting and gasping for breath the whole way.”

not untrue, but only part of the story.

certainly your math skill level neither makes you "smart" or "dumb" (which really aren't opposites, either).

prerequisites are (ahem) required. not having them does imply having a bad time.

what's missing is that different people's brains work differently and people have different talents.

if you learn differently, that can factor into that lack of prerequisite knowledge - perhaps the way it was taught didn't work for you.

but some people's brains just don't like math. other's are gifted at it. you can have all the prerequisite knowledge needed, be the best most diligent student, be wildly intelligent in general, and still not just "get" math.

so this article was about someone who actually did have a decent proclivity to math, but was robbed of it because of some missing foundation. and then said "ah ha!" there's the problem! but that doesn't mean that's the case for everyone else - far from it.

it smacks of the "affirming the consequent" fallacy:

    ("dumb" => !math) !=> (!math => "dumb")
    (!prereq => !math) !=> (prereq => math)

No, this isn’t universally true. Your intelligence does affect your ability to learn math. It is not always just a lack of prerequisites.

Most people are of average intelligence. Suppose that

-a few IQ points here or there makes little difference in one's aptitude in real-world tasks

-we then must accept that when most people think they are "dumb", there is some other effect going on such as:

-lack of resources, hunger, mental distractions, illness, or motivating incentives.

People often, but not always, lack the prerequisites because they weren't able to learn them when it was taught in school. And that was because they lacked the pre-pre-requisites because they in turn didn't pick those up when that was the material being taught.

In math, things build upon previous things to much greater degree than in other subjects. If you get off track once, it's hard to catch up.

But if you lack prerequisites because it was never taught in high svhool etc, that's a failure of the curriculum.

The way I learned programming was to start with the absolute most basic thing I could think of - changing the color of a button on click in Javascript. With that I could start doing slightly more complicated things, until I could get a whole job as a professional software engineer.

Learning is like climbing a staircase, and what you have to realize is you can't skip steps.

As an educator, this is the thing I always say. If you try to teach someone programming for example, try to make a honest list of all required prior knowledge. This is usually stuff that is totally obvious to anybody in the field, but if you don't know it it might give you a hard time. E.g. that programs run within the context of an operating system and the OS provides interfaces for interaction with hardware.

Not all of that is needed upfront, but certain explanations just won't make any sense if required knowledge is missing.

Related to this, it is why when writing documentation for something, it can be extremely useful to also list the prerequisites someone needs to pick up the information in the document before them. Possibly with also linking to resources about them, depending on your audience.

Not only that, writing out the list of prerequisites also helps the author write a better document. Because thinking about what knowledge is required serves the same function as thinking about a good unit test does. It makes you stop to consider "the obvious" and sometimes realize you have overlooked something.

Because when you are thinking about these prerequisites, you are likely also thinking about why they are needed and what challenges come with them. This in turn might lead you to revise aspects of the documentation to make them clear as well.

But how could some 7 year olds have vastly different ”pre-requisites” than others?

In my experience aptitude plays a far bigger role. Yes, you compensate for lack of aptitude with a lot of hard work, but that’s a different matter.

Smarter people will get the prerequisites faster. Nothing wrong with that, I'm more of a persistent person myself, which is a quality in itself. The really accomplished people are both smart and persistent.

But otherwise I agree with the article. I have zero basics in physics because my first teacher was generally senile and there was noone else (small town), and it was always something where I automatically tried my best to just get a passing grade.

Some courses in my university were restricted to students in their third year or above because of "mathematical maturity", which I thought was complete BS because some of the courses had no other prerequisites. But after taking some of them, I get it. There's a general sense of problem solving flow that takes time to develop.

Some credit needs to be given for just jumping in. Just analytically breaking down complex problems into pieces that are understandable.

It's often faster to work top-down and turn unknown unknowns -> known unknowns -> known knowns.

My step-mom tutors. A high schooler was having a terrible time in algebra. She quickly realized there was foundational numeracy missing. The next lesson they walked out to the residential street and asked: how many tires are there. The kid froze and guessed. Time to start counting. A couple months later, the kid was able to raise their grade to passing in algebra

This article makes me think of the skill tree or roadmap posts we’ve seen here on HN or Reddit over the years. For example https://roadmap.sh/frontend …

I feel like we could do a better job of providing ourselves fundamental tools like this in helping ourselves and others learn. Not just in tech, but in life overall. The “dev” tree above is embedded in the life skill tree that should start in elementary school.

Haha, even the life skill tree has “fictional” branches that intercept the game world skill tree … you really do have to learn all of the dependencies necessary to case 5th level fireballs … there are real rules to be learned in the games their usefulness is just siloed into the fictional realm.

Edit: I forgot to point out that roadmap is open source here: https://github.com/kamranahmedse/developer-roadmap

I’m curious about what that “preliminary knowledge” is? I’ve read stuff like a mathematician delight and the Joy of X and it’s such a beautiful, attractive but seemingly unattainable realm of knowledge.

As an example, this is the math that I’m aware of and have been exposed to:

Arithmetic Algebra Geometry Trigonometry Calculus

I’m vaguely aware of linear algebra but haven’t studied it (it also seemed unattainable)

I’m also aware of discrete mathematics and even bought the book concrete mathematics by Knuth, only to be totally stuck in the very first example of recursion and the tower of Hanoi…

So, what is that preliminary knowledge and how does one goes about acquiring it?

From where I sit sometime it feels like I don’t what I don’t know and I don’t even know how to ask how to learn what I don’t know I don’t know

This is an optimistic take on things. With years you understand there is a small cohort that is not capable of learning the basics even if they try, then there are folks who do not even know they are lacking.

This is a half truth. Part of the reason some people take a lot longer to grok the prerequisites and get left behind in class is because of cognitive ability.

Working memory (WAIS) digit span, and broader performance IQ (as opposed to verbal IQ), generally indicates how many conceptual 'items' you can have in your head at once. With more advanced math, this becomes _critical_ to forming the coherent plumbing between concepts in your head, leading to understanding.

Incidently, ADHD is largely an expression of specific personality traits and low working memory.

This is where LLM's have been the most helpful for me. When I am engaging with an entirely new subject and have a bunch of questions I need to pepper someone with, I can ask as many clarifying follow ups as I need without getting self conscious or worry about annoying whoever I'm speaking with. The LLM is infinitely patient and able to easily handle beginner level prereq questions.

This tracks pretty well with my experiences learning programming, both when developing websites and making video game mods.

The times I failed, I was looking at other people's work and trying to figure things out too quickly and in an unstructured way. I saw the complexities of a program that was in development for weeks/months/years, then basically panicked and thought I'd never be able to make something like that.

When I learnt the basics, I then saw how these problems could be broken down into their simplest forms, and ended up learning a lot more efficiently as a result.

Of course, having examples of what to do helps a lot, it's just your examples need to be merely a tad more complex than what you already know, not a masterpiece from some genius that spent the last decade working on it. Or if they are from that sort of person/company, you should try and break down sections of the work at a time to understand where they're coming from, not the whole thing at once.

It's much more reasonable to try and figure out how someone like Facebook or Netflix implemented a profile page or edit button than say, how the whole system works on a greater level.

You know how there's a window for learning to speak?

https://en.wikipedia.org/wiki/Critical_period_hypothesis

I posit there's a similar window for highly abstract thinking, like math or logical thinking or, controversially, for learning how to learn.

I suspect this highlights are more serious issue which is that most of our training methods are not adaptive. They work well only if the students arrives at the right phase in their understanding and otherwise make poor use of everyone's time. Yet assuming the student has something to learn, and the teacher knows about it, this does not have to be the case.

One discussion of training I found eye opening was Pat McNamara's thoughts on what I believe he said was called "skills-based training" versus "performance-based training". With skills-based training, instructor start out the training session with the idea in mind to cover certain skills. A lesson is successful if it covers the skills the instructor wanted to cover. Performance-based training is geared towards improving the students' performance, so skills are introduced based on the students' actual level of ability and the relevance of training in a particular skill for improving their performance.

One motivation for adopting performance-based training is the lack of success of skills-based training in many contexts. Why is skills-based training sometimes unsuccessful? One reason is that the skills may be too hard -- the instructor chooses the skills with imperfect information on the students' level, and they choose the wrong skills. The students receive the training but their abilities do not actually improve; they don't know what's going on. Another reason can be that the skills are too easy -- the students receive the training and actually meet all the standards, but it doesn't actually help them get better.

Pat McNamara discusses these concepts in the context of being a shooting instructor for police departments and military units. It seems that one often doesn't know what these units know before one shows up, and the officers and soldiers in any one unit can be quite different individually, so the instructor has frequent occasion to think about the relationship between what they planned to teach and what they actually did when prompted by the students' questions and challenges.

For many practical applications, slow learners do just as well as fast ones, once they’re up to speed.

Math teaching is mostly playing hide-the-ball, which teachers justify by saying people learn more deeply when they figure it out for themselves. But really that just shifts the burden of backfilling prerequisites to the student.

A lot of people hate rote learning - but elementary math classes, meaning all the way up to calculus (or prior to more rigorous proof based classes), do require a bunch of memorization. That's on top of getting an intuition...the "Aha!" moments.

I've observed that many math students in those types of elementary classes struggle because they're unable to recognize identities. They get some problem which involves substituting hard parts with easier parts using identities, but don't recognize them. So they try to solve the problem directly, and end up writing pages upon pages, before either getting stuck or doing some error that follows them until they get stuck.

Once they're showed what identities to use, they say "of course, I should have known that!" - but they never put in the time to solve all the problems.

And I was like that, too. I always thought that math would be a nifty because you'd "only" need to learn the various theorems, and if you understood those, then that should have been enough. It didn't really hit me that I'd need to put in hard work solving problem sets until I started recognizing patterns and knowing what to use, and where to use it.

Same goes for those that don't really understand the theory. Lots of math problems later will be of the type "Here's a difficult looking problem, is [statement x] true or false?" - and because they don't understand what math theorem to use, and all its properties, they'll try to brute force it by jumping into calculations.

You see it all the time in calculus, where students are asked to solve some nasty looking integral problem, which is much simpler if you know and use properties regarding symmetry and stuff like that.

I'd say for most people, there's no free lunch when learning math. You'll need to understand it, and you'll have to practice.

There's always going to be some extremely high-IQ individuals that can do pretty advanced math purely by logical deduction - but for the vast majority, it comes down to hard work.

I have still not graduated and this is my sixth year at university (in Europe). I find math too difficult and I still have calculus, linear algebra and probability. Discrete structures is the only math class I managed to pass and it took me too long to realize it's because the other classes have a long list of prerequisites. I have passed all of my software/IT classes aside from the maths and it's because they virtually are built from the ground up. On the other hand, strong math foundations are required even for the introductory math in my college. What I did was get great algebra and precalculus textbooks and went through them with great detail. After that I found the classes were not that hard to grasp.

The fact that the push primary school students through to higher and higher levels of math, when they 'fail', is a sacrilege to the student and a stain on the education system. I see so many capable students that don't have the prerequisites for the current math level, and are now just completely lost with NO chance of finding the path in the current system. And completely demoralized and disenfranchised.... all for what? To meet some BS metrics? This really needs to change.

That's one of L. Ron Hubbard's barriers to learning, described in his "basic study manual" book.

-Another one is not fully understanding the words or concepts being used.

-Another is not having an appropriate example or visualization of what is being explained.

I learned this as a teenager when I went from great at math to terrible because I got stuck with crappy teachers. Then, in 11th grade, I got put in algebra 2 with a great teacher and was tutoring other kids.

Math is completely different than other subjects. You can't catch up by cramming or reading a book over the weekend. You have to consistently learn and use it over the years. And have competent teachers to teach it to you.

Once you get placed in the remedial math, where they are just corralling misbehaving teenagers, and slapping out worksheets so kids can pass, you are basically screwed, unless you can get out of that situation.

Reminded me of the Feynman’s technique. I relate completely. One of my biggest challenges in returning to university after several years of work was exactly having lost the grasp of prerequisite knowledge. Unfortunately, from experience, more often than not lecturers just play the “you should know this from previous courses/high school” card and you are pretty much left alone in your struggles. Gets even worse if an exam problem relies on some borderline trick that wasn’t practiced throughout the course. You could probably tell I haven’t let go of some grudges.

This was my takeaway from college. In HS, you don’t have a lot of prerequisites, excepting the “II” level classes. I quickly found out how unstable my math and physics foundations were. Luckily, few realize that Bs are closer to failing than they are to mastery.

Even the college gut courses have hidden dependencies. I still feel for the business majors in my entry level stats class when the prof, bragging about learning calculus at 10, required calculus proofs for all the things.

Much like Civilization and Diablo games, and @godber’s comment, those tech trees should be required for all course syllabuses.

Late to the thread, but my view on being "dumb" in general, even if you have the pre-req's knowledge and ability are different things. Most people, in my opinion, are smart enough to understand and apply just about any complex subject or topic. Being smarter just means you can "compute" and understand or apply the subject in question faster than others.

In the end, what we prioritize and how much time is available for us to tackle different subjects is the biggest limitation, not genetics or luck. Art and entertainment heavily influence these things.

Prerequisite is a euphemism for practice. You lack practice.

It’s like saying you can improve your skills in basketball/swimming/piano/singing if you just practice better.

But obviously you can still be dumb and know a lot of math.

> A cage of 5 mice costs ~$1k upfront and ~$5k/yr recurring

You can get mice a lot cheaper than that, I'm not sure what kind of mice he's referring to but the prices depend on the vendor and mouse type.

Where I work it's about $2 a day to house a cage of 5 mice. It's about $30 a mouse if you get C57BL/6NJ's from Jackson: https://www.jax.org/strain/005304

So more like $150 for 5 mice and $800 to house for a year.

Another good one to know if the size of antibodies (10-12 nm).

The author's point about Elden Ring is especially on point..

Because not only do you need to be level 50, but you need to try and fail five times before you see any kind of success.

Failure is _inevitable_. Quitting is optional.

You have to learn from each mistake.

I discovered this myself when I attended a data science summer school, a 2-day bootcamp. I knew Python and Jupyter. I know basics of Operations Research, though I barely passed the class back in time. I took classes on optimization classes thanks to industrial engineering classes. But at the end of the boot camp, I was as illiterate on data science as I was at the beginning. I was just more confused. Then, I understood that I was missing the mathematical prerequisites for the understanding. I still felt dumb though.

I don’t even have to read the entire article for the title to resonate with me.

But when you are poor and really need a leg up in society, you will do anything to push yourself forward - including going into student loan debt.

I certainly wasn’t equipped nor ready for computer science. Well let’s say my computer science classes I did well. It was the Calculus and Physics that I struggled because I didn’t have a good background from High School.

I didn’t have the necessary pre requisites.

When I recently completed my Masters in Systems Engineering, getting a 4.0 GPA was no problem.

Assessments are getting better in education and they help find missed skills. It's possible the author was smart enough to copy but not understand why they were doing what they were doing. Whether your local district focuses on skill acquisition versus graduation rate may determine the students success long term. I know little, but what I've seen from reading programs they're pretty much 'there' in discovering these blind spots. I don't know if there is comparable assessments or programs for mathematics.

Lately I've begun thinking that the way maths is taught, with each new concept following on the previous and no real way to revisit older content, just might be why people think there is a sharp divide between people who seem to understand all maths immediately and people who don't get it at all. Of course it might be the case that maths people exist, but maybe it's mostly survivor bias.

It's absolutely true. At work I keep a notepad open where I write down questions to get to the bottom of things. What occasionally stops me is shame that I might be asking my peers too silly questions as expected from my level. It's a daily struggle but there's no other way for me. Life's about learning and growing so you better get the right mindset for it.

me except I’m socially dumb because I didn’t learn how to have friends in middle school

To generalize. When people fail they can blame themselves or can examine how to change. I saw one person try open a gate, fail and say they were no good with mechanical things. I said "Its not you, the gates broken" and the person immediately opened the gate. This is pervasive. "I'll see it when I believe it" as the saying goes.

Many people are in fact dumb, and will never acquire the prerequisites. We're not all blank slates with equal potential.

Math is hard but fortunately programming is easy. I am just a dumb soldier who taught themselves to program while traveling around Afghanistan.

The down side of being a dumb soldier programmer is that it’s really really hard to find sympathy when people complain about how hard life is when they are utterly reliant on a bunch of abstractions and clutter to do their jobs for them.

I can relate - I was quite bad at math through high school.

I eventually hit a wall in college then, like the author, decided to start from the complete basics: positive and negative numbers, fractions, arithmetic, algebra, then calculus.

Khan academy made this possible for me (in 2010), I don’t know where I would be without it.

The real truth: if you aren't good, there is nothing wrong with that and there are more than enough developers in the world and people who are good with math. What we need is more people creating real and interesting jobs for these skills.

Also most people aren't great with spatial reasoning. Chess requires zero prerequisites yet the average level of chess on chess.com is constant one turn blunders. It took only a year of playing on and off to get to 98th percentile and up to maybe 70th percentile most of it is capitalising on basic mistakes. We need to stop deluding people with feels good content, that's how you get memes like imposter syndrome.

When machine learning went mainstream I realized I didn't _really_ understand linear algebra.

That is what I dislike about Computer Science course ( BSc ) and much prefer Computer Engineering ( BEng ). There are far too many abstraction involves that most people just remember or know the skills set of the abstraction layer without ever understanding how that abstraction comes into the field in the first place.

Over the past 10 years the media has have popularised the term First Principle often spoke about by Elon Musk ( He didn't invent the term but media help to spread it ). And this is precisely it.

And this isn't just computer but literally every single subject taught are now about the grade and not the "WHY". We just dont know how most things are derived from. We just memorise it and society will reward you with Certificate and a "Smart" status.

In Maths Richard Feynman [1] explaining mathematics in 4 pages from algebra to calculus. As the saying goes, I dont have time to write you a short letter, So I wrote a long one. Getting something simple and concise in 4 pages is the work of genius and takes a lot of time. I only wish something like this exists for all other subject with video course, completely free of charge in dozens of languages to kids all around the world.

[1] https://www.feynmanlectures.caltech.edu/I_22.html

I think intelligence (however defined) is important. But the problem is often people misuse this metric to predict a binary conclusion of whether they can acquire a skill or not instead of just considering it as one variable involved in the learning curve.

That is why finding the right study materials for the fundamentals of a subject is so important. Take some time to find out the right method and material when learning something new till it speaks and inspires you. You will learn much better and faster.

This post biases way too hard into the nurture side of the equation. The difference between the author and someone who is genuinely smart is that the genuinely smart don't need to spend months carefully working through all those prerequisites in carefully arranged order. Until you meet somebody like this, it's easy to delude yourself into assigning yourself more brilliance than you posses and think that everybody struggles the same way. It turns out some people really are smarter than you, prerequisites or not.

It’s heartwarming to read comments from clever people, focusing on their struggles. Too often I interact with people who lock conversations into their own sphere of competence with the outcome being that I feel incompetent.

I’m enjoying going through your site, looking for inspiration and tactics. Btw, this page is missing: https://lelouch.dev/roadmap

but when I read a paper, it's difficult to know what the Prerequisites are.

Anyone here with developmental dyscalculia managed to overcome it and get into math? I have no clue how to deal with numbers, I know this is weird to say but it feels like they don't exist for me.

Math as taught by one of my teachers made a lot of sense.

Some topics will come easier and click. Others will need to be brute forced by practicing examples.

I can see how that generates pre-requisite knowledge one way or the other.

This is me diving into leetcode without majoring in CS in undergrad.

There's are also a bunch of precalculus stuff that comes in handy that I completely forgot. Like how to compute arithmetic sums!

The source of about 30% of the hits I took in homework and tests was, to this day, not having memorized the basic values of Sin/Cos(pi, pi/2, pi/4).

I would be curious to hear what “better learning methods” the author used. He calls them out but doesn’t describe them.

I have a corollary: people who are smarter than I am are mostly just less lazy than I am.

I am confused: how exactly did the author managed to lack prerequisites for math in school?

I am not sure I entirely agree with the premise: eg. you maybe are "dumb" (lack mathematical talent, really), but with proper instruction, you can learn a lot of math.

Let me dive deeper.

Our school system teaches math in a pretty inflexible way: "this is how everyone can get it". But even math talents don't learn it that way: as one, I was usually ahead of the school with my own reasoning (sometimes by a couple of grades) and could backtrack to the school method to understand it and apply it.

Second, if you are good at maths naturally, everything else at school becomes easier: people simply treat you as "smart" in whatever you do just because you have a natural leaning to mathematics (both if they do or don't themselves). Even rote memorization subjects like history and geography become easier since, well, you are "smart": teachers simply do not ask much of you.

And finally, I've met many an extremelly intelligent mathematician (uni professors and math competitors) who simply are outright dumb: they could not process a simple logic statement in human language, even if they were regularly working on advanced research calculus.

So, anyone can learn a lot of math, and doing so requires internalizing the foundations. However, people talented for mathematics find it easy to internalize them in various ways (not always the textbook way), so it's not hard work for them (eg. I could coast through the entire undergrad math and CS program too, cramming for a weekend for all but a couple of exams: memorizing all the axioms and theorems was the struggle, operating with them and proving them once I knew them was comparatively easy and I finished with a GPA equivalent of 3.4 or so).

But math instruction is hard because math is a formal language representing a very specific mindset that not everybody can naturally get. And instruction is usually performed by people not having attained that internalized knowledge of the foundations, thus not being able to look at it and describe it from numerous viewpoints required for individual students.

Finally, we need to fix the society not to equate "good at maths" with "being smart": plenty of smart people who have a hard time with maths, and plenty of math wizards who are outright dumb.

Books and courses usually list their prerequisites. Or good ones do, anyway.

Prerequisites are so important, and math is one of the things where it makes more of a difference.

>You Are Not Dumb, You Just Lack the Prerequisites

I know what you mean, after years of study I now feel confident that I don't lack the prerequisites to be as dumb as I could possibly want to be ;)

So many times the solution is to be kinder to yourself. I love this.

This page needs to get reshared 8 billion times by 8 billion humans.

How to tackle it?

To me there are either two ways: when you are trying to learn the thing XYZ you are seeking, drill down to the first thing you don’t understand and consult a lower level resource. Continue until you reach a level you understand. And the second way is: Re-learning “all” of essential math and then going back to XYZ.

I don’t think the second step is feasible, as you cannot possibly learn everything in a breadth-first kind of way until you are deep enough to learn the (now level-adjacent) topic XYZ.

But for strategy 1, the question is 1) how to identify the problem that you are lacking (e.g. how to isolate math gibberish into a concrete concept) and 2) how to find a good resource to learn and practice this concept at this level?

I do struggle with this and sometimes randomly learn some lower concept again but notice later it did not help me in the end and just left me with a million untied knots that were infeasible for me to entangle.

I wish someone had told me this in school.

Would you like to recommend any resource?

True, but then there's nothing wrong with being dumb. I know a lot of smart people, and they're all dumb about most things. Like the universe, we're mostly empty, with some hot, bright spots. What I mean is, don't think of yourself as fundamentally smart or dumb, think of yourself as having a lot to learn, no matter who you are or how others perceive you.

But sure, this is a good reminder of how you go about learning new things. It's the Julie Andrews method of pedagogy: "start at the very beginning (a very good place to start)"

I wish we didn’t think of understanding math as being smart or dumb. Like, it was a singular ability. Math is a subject deep, wide, and rich of ideas as well as results. For me, learning and understanding new concepts and results is like discovering new mountains and trying to ascend them. I won’t be able to ascend most, but I can certainly sit back and appreciate and enjoy their magnificence.

> It’s like trying to defeat a Elden Ring boss… at level 1. Just in love with this comparison

As a side note, the headline seemed a fun one, it seemed to me to say “you are not dumb, but you just lack a few prerequisites to be dumb”.

Intelligence is far too complex to be meaningfully described with a single number like IQ. Measures of physical capability don't suffer from this issue - a person might be strong, or they might be fast, and everyone knows that the power lifter and the marathon runner use wildly different training regimens to improve their abilities.

If physical capabilities are highly trainable, up to some genetic limit that the vast majority of people never even get close to, then it seems that intelligence must work the same way - e.g. prodigious feats of memorization can be achieved via training regimes (memory palaces etc.), as can one's three-dimensional visualization skills (e.g. a chessboard layout, or rotating a platonic solid, etc.) or the ability to rapidly construct arguments using logic and reason - but we don't seem to be able to classify different areas of mental ability as easily as with physical abilities.

Sadly, this is one of those politically difficult topics as the blank slatists and the genetic determinists (Lysenko vs. Galton) have tried to use all kinds of pseudoscience to support their ideological arguments, when the underlying point is just that training your mind is as beneficial as training your body, and everyone should do it at least to some extent.

Yeah but good luck going through the entirety of khan academy as an adult, it's not impossible but it's also a task as hard as learning a new (or multiple) languages...

If you can dol it though and complete everything up to precalc you can most definitely do well in university.

Hehe, two years ago, I wrote a similarly titled article - "You're not dumb, the prerequisites are bullshit." :) https://news.ycombinator.com/item?id=30035456

I often tell people they're not dumb (and therefore I'm not smart), they just lack the prerequisites. But they never go and acquire said prerequisites. I do. Maybe they are dumb.

> This belief shadowed me for years, a constant reminder that while believe I am smart… I’m not THAT smart.

Sentence missing an I

> It’s like trying to defeat a Elden Ring boss… at level 1.

an

> In fact, I’m still pretty dumb.

Contradicts the first sentence I'm quoting.

Streams of tears roll down my cheek as I write this because this article perfectly highlighted that it wasn't the laziness but rather what was causing it , mainly the lack of prerequisite fundamentals needed to thrive in math field. Had I known this my life trajectory would've been different instead of self loathing and inferiority complex I built up around something so innocently simple.

The rush of epiphany and self-forgiveness that overwhelms me after all these years. I realize now that learning grade school math in French and then started to learning algebra and calculus in Japanese abruptly moving to an English speaking institution to continue math degree (which i abandoned for reasons in the article i realize now ) screwed me up big time because neither French nor Japanese nor English is my first language.

For instance I would store numbers in French in my head and perform arithmetic in French but to do any sort of additional algebraic or calculus I would need to switch to Japanese internally and finally write out response in English. Learning the advanced topics in English was never going to work out, it was like building a castle on sand and the stones are made out of mud.

     I always thought I was too “dumb” to understand math. During my school years, it was evident to me that for some kids math was easy, and for others like myself: painfully difficult.

     This belief shadowed me for years, a constant reminder that while believe I am smart… I’m not THAT smart.

     Recently, after 150 days immersed in learning math, I had a stark realization.

The struggle wasn’t because I wasn’t capable, but rather, I was simply missing a shit-ton of pre-requisite knowledge.

I wish I could show this article and translate it into other languages. There are lot of young kids in schools who tell themselves they are dumb or lazy because they can't do well in math and sciences.

God knows how many of us are walking around feeling inadequate or frustrated at ourselves because we convinced ourselves we are not worth it or capable when in reality its the prerequisites both conscious and subconscious, overt and covert we fail to realize as fundamental stepping stones to success.

It might as well be that failure in startups or business ventures or relationships even also stem from this principle: that the fundamental prerequisites were not taught or caught early on (either due to environment, upringing, socioeconomic constraints) have solidified into bad habits, bad model of world, bad model of others that ultimately transpire into bad thoughts, bad words, bad actions and opposite outcomes of what we set out to accomplish.

Going forward I must make it my mission to realize what fundamentals and prerequisites I do not have and instead of brute forcing and letting my ego ignore it, I have to put aside time to build those basic building blocks.

A cathartic angst feels deep in me. Might be too late for me due to my age and I fear I will ignore my own writing here and others will too. It's truly sad that we are all realizing it this late and will forget whatever lessons were learned. I wish society and people would stop pointing fingers at people and rather realize build tolerance from the fact that not everybody gets to build the same prerequisites as humans cannot be the same, some are innately inclined to better at certain things while others are not.

Equal outcomes is a failure in the making and schools need to stop and focus on helping students build prerequisites on their own schedule and pace.