You are not dumb, you just lack the prerequisites

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.

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.

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.

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.

> “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.”

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)

-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.

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.

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

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

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.

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.

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.

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

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

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

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.

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.

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.

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.

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.

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.

-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.

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.

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.

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.

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.

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).

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.

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.

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 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.

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.

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

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.

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

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.

>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 ;)

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.

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)"

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.

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

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.

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.

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.