Your previous education, your upbringing, your cultural values, these things all also have huge effects on your aptitudes, and you've just dismissed them out of hand, apparently in favour of pre-determined genetic intelligence. I mean, all I know of you is this comment, so I could easily be missing a lot more context about this argument you've had, but it sounds like you have an axe to grind, not a carefully-considered conclusion.

To start, I believe that the idea of "working memory" is largely valid. Think of it as the number of distinct ideas you can hold in your head at once, sort of like trying to hold a phone number in your head when you've just heard it for the first time.

The general consensus in psych is that this number for the average person is in the single digits and is relatively static in adulthood till a decline in old age. It's been my observation that people with really incredibly small working memory cannot do algebra. The amount of numbers/ideas held in their head is too large, and multiple students in this group described the experience of attempting an algebra problem as feeling like sand constantly slipping through their fingers.

Many of these students grew up in rich neighborhoods with good parents. They had most advantages you can imagine, seem reasonably intelligent when you talk to them, but Algebra will always be beyond them.

I can get around this by brute forcing the numbers into the formulas on paper and going through each step slowly but surely, even steps that people can do mentally. I just need significantly more time than my peers to finish.

Taking stimulants and increasing my dopamine levels across my synapses alleviates much of the working memory deficits I outlined.

I wonder how many of these students that struggle were just undiagnosed ADHD and not deficient in an low IQ mental handicap sense.

I've got ADHD that was undiagnosed until a decade after finishing my degree, and maths was my favourite - and best - subject all the way from age 4 through to doing solutions to Einstein's equations involving 10+ A4 pages (both sides) of tensor mechanics for my final year project. My results across that time showed the more maths in a subject the better I did, the more writing it had the worse I did.

Doing maths is like reading is for me - an external cognitive structure that I can follow to make my own brain calm down while in that process. While I don't do much maths nowadays, I literally read whenever I'm not actively doing something else - I read a page or two of my book in-between clicking reply to your comment and starting typing this reply.

Maths involves a lot of "muscle memory" once you get past the initial hump. But it's often poorly taught at an early age to the level of inducing near-phobic levels of discomfort with it which to me seems strange. But I don't think there's any extra issue with having ADHD and maths

ADHD is very comorbid with dyslexia, dyspraxia, dysgraphia and dyscalculia - around 20–60% of people with ADHD also have one or more learning difficulties. Dyscalculia affects as many people as dyslexia, but dyslexia is far more well-known and more likely to get diagnosed and helped with. There's a nice summary box of typical symptoms of dyscalculia in this paper:

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6440373/

Interestingly, testing myself with the Wechsler reverse-digit-span test, I found very large improvements in working memory from taking 20-minute afternoon naps and from modafinil. I'm not claiming everyone is alike, of course, but working memory is definitely not as fixed as height and eye color.

Specifically with regard to mental math, I find I can do a lot more when I'm lying in bed in a dark room than when I'm in an uncomfortable chair in a noisy cafe with people talking to me. Or, for that matter, classroom.

> Many of these students grew up in rich neighborhoods with good parents. They had most advantages you can imagine, seem reasonably intelligent when you talk to them, but Algebra will always be beyond them.

I tried marijuana once and found the opposite effect: when I was high, by the time I got to the end of saying a sentence, I couldn't remember how it had begun. But people report that I seemed like I was conversing normally; if they didn't know me, they wouldn't have realized anything was off. I wonder if these folks were experiencing something similar all their lives?

Sure. My ability to do anything also drops to near zero if I don't sleep enough. Yet I have not found unlimited increases in capability if sleep more and more...

> modafinil

Yeah, drugs are a quite different beast. Doping happens when individual athlete realizes they have hit the limits what their "natural" biology can do and yet still are not going to win the competition.

If you are functioning at the optimal point - stimulants will only make your performance worse.

You are likely sleep deprived, for whatever reason. That's all.

Wait, are there mainstream schools of thought where working memory isn't considered valid? I'm a layman but I suffer from ADHD, and very much notice that my working memory fluctuates with my attention span (from lack of sleep/stress/etc).

Google's of no help to me, but I remember a story of some educators, looking at some kids who didnt go to school but worked selling concessions, but in turn, were actually quite good at math. They couldn't answer math questions when written out on a worksheet, but they could do the exact same questions when presented in the form of a complex order. (Double digit multiplication and summation isn't the same as algebra, but being able to do that implies a large working memory, which is claimed is the barrier to learning algebra.)

But I did make paper sort of my working memory and when doing algebra I felt that I was just the very resource limited CPU that executed the instructions from paper-memory.

Algebra always made me feel like I was doing some mindfull exercise where I had to empty my mind, follow the paper script and hope I didn't mess anything while switching from row to row of calculations.

Even today, as a programmer, I struggle to remember class or function names, I just empty my mind and am really good at searching stuff in code.

> There is also evidence of domain-specific cognitive deficits that contribute to specific learning-related disabilities. For example, phonological processing difficulties have been found in children with poor reading performance, whether or not they also exhibited problems with ADHD symptoms or math, but not in children with deficits in ADHD symptoms or math only. Similarly, both with and without a reading deficit, children with ADHD symptoms exhibit significantly impaired object naming and behavioral inhibition, and math-disabled groups demonstrate visuospatial and numerical processing deficits, while those with only reading problems sometimes do not.

There's significant co-morbidity between learning disorders and ADHD, and despite dyscalculia being as common as dyslexia (~3-7% of the population) it's a lot less well-known, isn't as frequently diagnosed and there are fewer tools to help people with it. It would be very possible to have both ADHA and dyscalculia, given you used symptoms listed for both that are almost word-for-word identical with those in the paper...

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7079676/

(paragraph 3 of the main discussion)

My problem was that oftentimes I would screw up calculations for a math problem and only get 7/10 score on it (the reasoning was mostly right, but the answer was not).

You sequenced the genomes of your students and correlated their performance against their genes?

I haven’t performed any genetics tests but brain composition is partly determined by genetics so genetics might play role, right?

I got the feeling as a kid that a lot of my classmates would just give up. I mean they're sitting there acting like they're concentrating on math, but they aren't. In other classes intermittent attention is enough, you pick up social cues and repeated stories that you've heard, and poof you have a history essay. With math there's a need to have all the steps.

How do you control for that as a teacher observing the kids? How do you know whether a kid has actually tried to learn stuff at home?

I still remember my brother was baffled at how I got top grades in math when he never saw me studying. Of course I tended to do that when he wasn't around.

The only solution I ever found was not having more than 3 students at a time, not an option for most teachers; I spent most of my career working with 1-3 students as a result. With that few students you can carefully observe the mistakes students are making and ask individual questions about their mental state. Experience will eventually tell you to differentiate students not putting in effort from students who are so lost that they're just flailing and hoping something sticks.

With more students, my experience was that both my attention became too split to give students the kind of careful diagnosis to control for effort.

I have. Some people are dumb.

You try teaching these people (anecdotally they're less than 1% of the population, with no obvious markers, so you have to actually find one first) and then come back and tell me how it's like "getting algebra to run on someone with less working memory". If I wanted to continue that analogy, I might say that the task is like getting algebra to run on someone whose brain has a power supply that randomly shorts out and spends half its time browned out.

Because some people out there are just not easy to teach.

A week of playing Kerbal Space programm gave me better understanding of orbits than years studying calculus, physics, general and special relativity.

It has poor applicability in everyday life - we learn all the biology of a cell but kid's don't know the difference between aspirin and ibuprophene.

But you can't base any argument on that when research into alternative (or complementary), genetics-based explanations is being stiffled. Or well you can, but that's just society-scale version of googling for statements that you agree with instead of questions that they're supposed to answer.

When you stiffle research directions for political reasons, you're doing politics, not science, and the arguments based on lack of stiffled research don't hold any more water than arguments based on no research.

Unfortunately, in practice science often turns out to be almost as much of a vehicle for confirming answers one already has in mind, rather than open-ended investigation.

I said absolutely nothing about what percent of people this is true for and absolutely nothing about why this is true. I brought it up in the context of the article because saying that not everyone can learn algebra is as taboo in education as the thought that genetics plays a role in poverty and success in life in psychology. My reason for thinking this is responses like yours.

There are clearly people who can’t learn algebra: intellectually disable people are such an example. I believe some level of “intelligence” is require to learn algebra.

How can you conclude from what I’ve written that I haven’t thought much about my belief? How can it appear I have an axe to grind from what I’ve written? You have formed an image of me that is wildly incorrect. Do you initially assume that everyone who thinks not everyone can learn algebra has an axe to grind?

Presuming that there are only mentally disabled and normal people is the weird hypothesis. The baseline idea should be that it varies just like health and height. Affected by both genetics and outside factors such as malnutrition and injuries.

Look closely enough and there is no essential difference between genetics and other causative factors. Other than maybe some people jumping to the conclusion that one has an axe to grind with minorities when one attempts to explain certain things with genetics. Which is just as much an arbitrary social taboo as the preceding taboos that constitute what we today call bigotry. (For the record, I'm a staunch opponent of all forms of violence and oppression.)

For me it makes exactly zero difference. Even if free will does exist in some essential sense, I do not believe that people generally choose what opinions to espouse. They simply acquire them through mimesis of their social environment. If that makes me a nihilist and a coward, then so be it.

Thought experiment: English Prime but also excluding any constructs expressing intentionality. I dream of a world where the concept of free will is considered just as poor taste as racial slurs. I think that, perhaps paradoxically, it will be a much more free and just world.

I like this idea of a "deterministic" language. In fact it reminds me of Nonviolent Communication, and is probably a good tactic for discussions that might otherwise devolve into personal attacks.

How is this any different from saying "all is one, separateness is an illusion"?

Which is exactly what you said, except that you chose to ignore that connotations conduct meaning, when you asked your rhetorical question. This is not e-prime, it is a plain old adverb answering the question "how?" like adverbs normally do.

My original comment had the purpose of questioning the validity of the "nature vs nurture" distinction. It just seems like an unhelpful distinction, but then again I'm not a biologist, just a lay person who likes their concepts tidy.

Genetics is obviously not the same as everything else in existence; I'm not sure that even makes sense as a statement. You seem to have somehow derived that I am arguing against the concept of distinctions at all. I'm not sure if language would be feasible without distinctions.

I don't disagree with any of what you just said, but I fail to see what point you are trying to make or what you are arguing against. Without the (linguistic) act of making distinctions, everything is indeed one and the same, but that's... kind of pointless?

EDIT: Sibling poster also seems to fail to make the distinction whether (a) we're comparing genetics to other causative factors, or (b) we are comparing my statement about genetics to your "all is one" interpretation of it.

In case it's still unclear, (a) and (b) are two completely separate things and I'm not sure how this conversation got to the point of conflating them. It just serves to reinforce my belief that the ambiguity of our language's syntactic structures makes it inordinately difficult to reason about many things in everyday language. Or maybe I'm just a bad communicator. "Me bad", "you bad" that's supremely easy to express lol

EDIT2: Correction, TheSpiceIsLife does actually get it.

The confusion is in the difference between proximal and ultimate causes, the rest of the discussion is over the ultimate cause of certain phenotypic features being down to genetics, some other mechanism or not significant at all. You've then said "all these different things are just proximate causes and [because free will doesn't exist] the ultimate cause is the laws of physics" to which people have unsurprisingly gone "what the hell does that have to do with anything?" because, well, it doesn't.

The fact that the ultimate cause of me taking a dump is "the laws of physics" doesn't mean the proximal cause wasn't me 'deciding' to go to the loo, and the fact that you can always say "the laws of physics" (or some higher power) is the cause doesn't make talking about higher level causes any less useful.

I don't believe in free will but its such a good trick that you act as though it were true almost 100% of the time, and talking about my 'decisions' as causes is useful the same as talking about 'genetic' and 'environmental' is useful. We talk about the causes of the Big Bang usefully despite time only coming into existence when the Big Bang happened :)

> How is this any different from saying "all is one, separateness is an illusion"?

> It's not arbitrary. "Other causative factors" is boundless.

These are distinctly different lines on inquiry. One is an inquiry to in the illusory nature of separateness, the other is an inquire in to the boundless nature of causative factors.

When we don’t even understand ourselves what hope is there we will understand other people? It’s definitely an ongoing process anyway.

Like another commenter said, operating with a comfortably skewed mental model doesn't help resolve the actual socioeconomic issues.

Good try at the "gotcha" though ;D

Previous education, upbringing, cultural values, etc are all separate effects that may influence your overall ability, but intelligence __definitely__ influences your ability.

Low intelligence + good education = poor overall ability

They didn't seem to do this at all. It can simultaneously be true that there are some who, due to genetics, simply cannot complete a particular task, no matter how conducive an environment they are put in, while for others genetically do not have these hurdles but whose success is still dependent on their environment.

If anything, you are the one dismissing this possibility out of hand, and making leaps of logic.

I have found with math, the way it's presented matters more than most subjects.

At that point people are trying to help you but saying things to you as if you are stupid, but you still don't understand. That really sucks, so people get afraid of math. Avoiding it, and nodding when asked "do you understand" when really they do not. This is hard to fix because you need to go back to the point they did not understand, but the fear and pain makes even teaching that a lot harder to fix as well.

My feeling is that most people should be able to understand algebra. However, I think that requires a very deliberate and personalized approach for some people. Certainly with collective classes, if you go at the speed of the slowest student there will be slow progress and a lot of people who are bored and mentally check-out.

If any approach is going to work for people who have real difficulties, it needs to be small-scale personal teaching, and it needs to come with trust. Someone needs to feel like they can keep saying "no I do not understand" without disapproval, disappointment, or frustration from the tutor.

I can definitely say that there are many people for whom it is a struggle - even those on college courses.

But a hundred years ago we could easily have the same conversation about simply reading and writing - they (the poor, women, or "lower class" ) luke not be taught to read. But it turns out that if you start young enough, and put enough effort in, 99% of everyone can learn

So, perhaps society is not putting enough effort in, or asking enough effort from, our kids for them all to learn calculus.

As the Agile Manifesto says, the work delivered represents the effort input so far. If society wants all children to learn calculus, we need to pay that price.

Edit: I think that we do need a new conversation about education. We have My father left full time education at 14, at 18 most of my peers left school and only 25% of us went to college. Today it's 50% it the total spend has not gone up in line.

I honestly don't know what kind of world the "universal education" advocates of the 1870s were imagining - but I am damn sure it was not people doing college courses on their iPhones, but it is the world they ushered in.

We need to double down on what worked for the 20th Century (and avoid the, y'know wars and genocide and stuff).

Education will lay at the heart of that.

[1] https://en.wikipedia.org/wiki/Ratatouille_(film)

There is a thing called functional illiteracy [1], where people can write and read but mostly only their name and some very basic things like grocery lists. They also cannot comprehend texts even if they can read most words. It's more or less equivalent to being able to add numbers, maybe multiply numbers 1-10 but very far from algebra.

No first world country can claim a 99% literacy rate unless you count in these people, which would stretch things quite a bit.

1: https://en.m.wikipedia.org/wiki/Functional_illiteracy

I would be surprised that a psychiatrist would find it a useful term - in the same way that any doctor would not look at the paralympics and see only one illness / disability but dozens of conditions.

I am fairly certain there are Rain Man like people who can do algebra or calculus but would for most people sit under the term "disabled".

Perhaps it might be good for you to ask, "the people I see who can do algebra also can act in certain other ways, socially and mathematically. How much am I looking for people who can do maths, people who can do maths like I do, and people like me who can do maths"

Edit: yes I suspect I sit to the left of you, but the important takeaway here is not that some people are more "intelligent" than others (I agree for whatever definition we choose) but that raising the education level of all in society massively benefits us all. Oh and ranking peoples value in society based on intelligence is not a good idea. It's only just above ranking on physical disability, or your parents aristocratic connections.

I gave no indications that people ought to be ranked in terms of value to society based on intelligence or any other criteria.

Their argument was purely an philosophical one, stating that a well informed society operates better, makes better decisions, ect. And a well educated person can make better decisions for themselves and have a higher quality of life.

The reason that universal education caught on was because large businesses realised that could have more efficient factories if they didn't have to teach everyone to read. Thus college and university because the place where the 'universal education' dream could be realised. Where every person could go and receive an education that would afford them a higher quality of life and allow them to better engage with the world.

Then, factories became more specialised and 90% of universities became slaves to some "industry"

Many universities "computer science" programs are hardly that. Rather they've turned into a coding bootcamp with slightly more math.

But there is a reason 2nd graders dont do book reviews of classic literature. Even though they can absolutely read the complex words out loud on autopilot any day of the week.

Edit: Actually, I think your premise is wrong. Children don't read words they don't understand "like muscle memory". The differences in meaning between, idk, betrothed, engaged, wedded and living in sin are superficial but they carry with them implications of social value, history and more. Children are supposed to read books and pick up context and understand meaning as they go along. We don't expect a 10 year old to grok calculus any more than we expect them to grok social/sexual arrangements. But we have different standards for 19 year olds - in the case of sexual arrangements it's obvious why. Perhaps less so for calculus.

Reading and comprehension levels are hard to guage for this reason - at a certain point the words are placeholders for political positions. And it's hard to tell if someone does not understand the word or disagrees with the politics. Please see every election for past ten years for reference

It's known that intelligence can be significantly negatively affected by environmental factors like malnutrition (both of child and mother during pregnancy), childhood disease, pollution and injury, and there is some evidence (also mentioned in the article we're discussing) that a century or two ago the average person was significantly dumber than now, presumably because of those factors - especially to the poor and women, which often experienced more severe childhood malnutrition than their male siblings.

So it would seem plausible that a hundred years ago a nontrivial percentage of people actually were too dumb to succeed in literacy and perhaps the fix for that wasn't just starting young enough and putting effort in, but rather that we started getting much fewer kids with severe environmental damage to their intelligence.

The richest kids at my highschool were also the smartest. There were one or two rich kids who didn't make honor society. There were also only a 5-6 smart kids who were middle/lower-class. Graduating class of ~500 in 1985, about 50 honor society (top 10%).

Doesn't this contribute to lower the reputation of the universities?

I mean, I did sociology in a small university in northern Spain and there was debate between the stats and methodology professors and pretty much everyone else about this.

In the end I'm grateful to my stats and methodology professors, because having to work very hard is the only thing that actually gave me something useful to go and fight with the world.

Many of my peers had to go to other unis because they couldn't pass there, so in the end they get a pass like me, the same in the eyes of employers, but they know it's bullshiting, and I know too.

IDK, It seems to me that there's a lot of lazyness in social sciences, and this kind of hard attitude to it is not only good for the students, but needed in the field.

IDK in the US, but this professors got a lot of heat for their stances on what a university is about, but they stood their ground, even when they've got hit by political bullshit, denied money for research, etc.

There's another thing (again, IDK if it's the same in the US), but why don't you see it in CS or Engineering faculties? I mean, I remember in my uni the students of this faculties had maybe a couple of professors that where ok to lower the bar a bit, but most of them just assumed that you gotta learn what you gotta learn. And many of them where pretty hostile to students, 180º stance from what I see in Stanford Moocs for example.

The way to make sure your students all pass is to do your job, those students need a teacher that cares and not one that 'needs a job', and preferably one that does not start off with the self-reinforcing viewpoint that a good fraction of them are too dumb to learn the material on offer.

Seriously, consider changing jobs to something with less of a negative effect on people's lives.

I've been on the receiving end of this attitude and it harmed me quite a bit, later on, with better teachers some of the damage got fixed but I most certainly would have traded two of my math teachers that 'needed a job' for some of the (much) better ones. Thanks to Fred Pach and one Mr. Groot I'm not a total loss at math and it ended up not being my skills but those of the teachers that were deficient.

It appears you are letting your past personal experiences distort your judgments about me. It’s ok. I understand this. It’s natural but it makes it hard to have a discussion about whether or not it’s the case that everyone can learn algebra.

You made a statement that the way to pass everyone is to do my job. Clearly you have not taught much in a classroom with a wide variety of students. Consider the possibility that you simply don’t know anything about the craft of teaching other than your very limited experience of being a student and that as such you should be wary of making negative conclusions about me.

I actually do because I read all of your comments in this thread.

When I mention that I just mostly pass my students now they understand that grades are an administrative aspect to the job and know that this is detached from my desire to get as many students as possible to learn as much as possible. People with extensive experience don’t harp on my statements about grading. At most they’ll claim they don’t lower their standards but they don’t make any wild conclusions about me like you did.

But you, you’ve made wholly unjustified conclusions about me and that says you don’t know the nuances involved. You don’t immediately understand that one can think not everyone can learn a topic and still be passionate about learning.

You’ve obviously not taught much in the classroom. Why is it so obvious? Consider the possibility that your thinking on this issue is clouded by your past.

Are the best Universities good because they have the best teaching, or are they just selecting the best people, who would be successfull no matter what?

The universities are penalised in rankings when a student fails to pass, but they are not penalised if they are so selective that the student never got a chance to study in the first place. This breeds an elitist system that does not let students rise to the challenge. Thats also one of the reasons why european universities do worse in rabking than UK ones - it's common there to start with a huge class, but only small part will co plete the course. In UK its harder to get in, but vast majorty seems to pass.

If universities worked like a normal business does, you'd expect that the best universities would grow, expand, and eventually there would an Oxford in every major city, with oxford teaching methods. But that's not how it works. But people still keep trying to appply 'free market theory' to education, when it obviously works very differently.

Lastly there is perpetual conflict between research and teaching - many professors want to do research and don't like teaching, many post graduate teaching assistants are folks that, no matter how briliant, were afraid of the job market, etc.

I think most people agree that there are extremes. There probably exists a small number of people who literally cannot (just like how 1 in a million could probably teach themselves calculus at age 10, also exists). However it's a big jump to conclude that even a small percentage of the people at your college cannot in the literal sense - i.e. if their life depended on it, they had all the resources they could want, they had no other distractions.

You just gave the definition of "cannot". Those circumstances will never happen, they're pure hypotheticals. Sure, hypothetically I might be able to play competitive tennis against Nadal. Practically, though, that'll never happen, I cannot and will never be able to do that - pretending otherwise is just naive and purely theoretical discussion.

Still though, we could ask how many people in the entire human population could run under a 4 minute mile with any amount of training, time and resources? Obviously more than have done it, but how many? Certainly not remotely close to everyone.

If the claim was merely that math comes easier to some than others - again it would be an uncontroversial claim and i would agree with it.

However, my reading of the original comment was that a good portion of the population literally are physically incapable under any circumstance, including the silly hypothetical circumstances i laid out, to learn algebra. I think that is false. It sounds like you do too. So i guess we're in agreement.

Not exactly. I read the original claim that they are "incapable to learn algebra even in very good/way above average (but still realistic) circumstances".

* What percentage of your students hate math?

* How do you differentiate between someone who doesn't get it because it's beyond their mental skill level versus someone who doesn't get it because they'd rather be doing literally anything else?

* When you say "mentally disabled", what exactly does that mean in the context of a spectrum of ability levels across the species?

* There are a lot of reasons people might fail your class. What are they? What's the percentage breakdown?

* What are the dangers of thinking that not all people can learn math? What are the dangers of not thinking that?

Here are some more thoughts to ponder:

Children are easier to brain wash than adults and teaching basic math involves a great deal of brain washing. To what extent is it the case that a person with no exposure to algebraic concepts growing up is at a disadvantage to learning algebra as an adult? There is a theory that if one doesn’t learn a language in childhood then it is impossible for them to learn one as an adult. Is something similar applicable to Mathematics?

I don't think the issue is necessarily that they're incapable, it's more that they don't have the necessary background intuition or knowledge. At least in the state i taught in, this is not surprising as elementary school teachers often dread math and fail their standardized teaching test math portion several times sometimes. It's the blind leading the blind.

I've also noticed it in reading, when they have teachers who aren't readers themselves trying to teach it to them.

Though I don't know of any studies about algebra in particular, kids show greater neuroplasticity in general than adults. So learning anything as an adult tends to be more difficult.

But I live with an American who reasonably fluent in Norwegian after 2 years of study in their 30s. It's certainly not impossible.

I have no doubt that math, like virtually everything, is more difficult for adults to pick up from scratch.

My conclusion is that some people simply can’t learn algebra. Be it genetic, environmental, lack of intellectual fortitude, lack of “intelligence”, or any combination thereof it is simply the case that some people can’t learn algebra.

Considering that a small percentage of careers require more than basic algebra/geometry/statistics, it's not hard to see that math has a "why should I learn this?" problem more than a "am I able to learn this?" problem.

However that feels really different from saying a not insignificant portion of the population, inherently cannot.

Not all people can read, it's obvious since the mentally disabled can't. Yet with proper education, all non-disabled people can be taught reading.

But that doesn't mean everyone can read, if you're illiterate by your 20s, you're gonna have a hard time catching up. Same for mathematics: most people reaching even high school are too mathematically illiterate to catch up[1]. Is genetics a factor: definitely, but it's among many others.

The reason why it's a partisan issue is the following: if I say genetics is a decisive factor, then I can say «it's natural, there's nothing we can do so we don't need to spend all that government money trying to help those people». The left-sided point of view goes as «There's nothing we can do about genetics, but we can change everything else. Then we need to find what are all the other factors, because those are the actionable ones». The conservative focus on genetics is mainly a justification for doing nothing.

> There’s a level of “intelligence” that’s necessary to learn a given topic. Not everyone can learn all topics.

“intelligence” is conveniently pretty ill-defined, but I don't think I'm more intelligent than my doctor friends, yet they struggled a lot to grasp even the most basic concepts of algebra when I tried to help them during our studies. “Not everyone can learn all topic” but I have yet to find evidences that your ability to learn a random topic you're not interested into is correlated with the common acceptance of the word “intelligence”.

[1]: that doesn't mean it's impossible, just likely well beyond the amount of effort they can (or want to) afford.

In some schools around the Bay Area, we have half the students on the fastest track, and about half the math faculty as calculus teachers. An immodest minority of students complete Calculus BC early and go on to the local college to continue their math education for their remaining years in HS. And these aren't even the top 50 schools in the nation.

The gains become clear depending on the degree to which you know what you want, and especially if you're thinking about graduate programs.

1. Calculus in some sense is somewhat disorganized, and it sucks that it's a gateway to more math in the US. By finishing Calculus early you can move on to the math you wanted to learn, such as Linear Algebra.

2. University students may have to spend about a year learning Calculus. That's a long time, and to some people they'd rather spend their money better by learning something else or graduating early.

3. If you're thinking about graduate studies in something technical such as Econ or Stats, then you'll probably want at least Analysis. The problem is that in the grand scheme of things, even Analysis does not leave you very prepared to do things, it just makes you prepared to learn more, so you may want to get ahead of this problem.

Because 95% of the time high school work doesn't transfer to the college level even if you do semesters at De Anza. You're back to scratch unless you go to flagship in-state, and even then you can only transfer up to a certain amount (at my undergrad it was up to Calc 2 - several people I know took Calc 3 in the same institution in high school but had to take it again).

Even if a college didn't want to transfer the credits for Calc 3, I'm surprised that they wouldn't allow you to skip the course. Also, for your friends, if they took a class at the same institution... doesn't that mean they got a repeat? Strange.

I also wouldn't generalize these things to people who occupy university posts, as they probably had interesting trajectories.

yes

> as they probably had interesting trajectories.

Not every physics professor went to a top school and published groundbreaking research.

I do want to emphasize that taking Calc 3 again is one class out of dozens you need to graduate.

It’s too bad, because I don’t think anyone should be able to leave high school without understanding compound interest-sorta vital for participating in a modern economy. Also, it’s not like we can stop loaning money to the innumerate, even if that might be the ethical thing to do.

If you could do that, you could probably pass Algebra II.

Context: Recently there was some controversy about Oregon graduation requirements [0] and social media represented the issue much differently than primary/secondary sources.

0: https://apnews.com/article/health-oregon-education-coronavir...

Of course, one may learn calculus from a book, which was what many of us did in that class.

In my country, highschool last three years. The first year, i was living in the dorms with some of my classmates. One, let's call him M, was fairly dissipated and, while everyone would rather play and discuss than study, and had shitty grades in everything but biology. Math was his weakest, with grade ranging from E- to D+ (i try to convert grades here, my grades wer in the A range and the average was B-), and English/spanish were not a lot better.

At the time, at the end of the first year, we had to choose specialities and which kind of diploma we could do. the "general" kind, with to specialities, economics/humanities or science, or the "technological" one, with three (electric, mechanic, civil). He realized at the end of the year that the only thing he wanted to do was biology, but as this subject was only available in the science speciality, he was fucked. Impossible to get to the "general"branch with his grade. He struck a deal with the school administration: 4 hour a week added to his curriculum, one hour every night after eating and before going in our dorms to make sure he worked. The year, in math, did not start well. He had a F/F- as his first grade (the average was D+, top of the class was B), but our professor was exceptionnal. we had 3 math illiterate in our class, and while the notation was pretty harsh on them, he took time to help everyone understand with interesting examples, stories and exercises. He even took hours of his own time for supplementary lessons. Our class relation was special too, we were only 20, and spent a lot of time together, even after college. Last time i talked to M he was doing a Master in marine biology, and aimed for a doctorate

End of the 3rd year, we have to get the national test. Our class ended up with an average grade of A-, M had a B+ and wasn't the lowest scorer in math (of our class). Miles ahead of the nationnal average (C+/B- that year). It was a public technological highscool in a rural area. The other close HS was one dedicated to farming. Culturally, we did not start at an advantage compared to the average kid of our country, i'd say we were at a disadvantage. But great teachers and small, but close class can help you emancipate from some of your determinations.

Another assumption is that the function is continuous. That's one I'm not sure about. I don't think there's a continuum between folks with mental disabilities and folks without. I think there's a discontinuity -- or perhaps overlapping spectrums on different dimensions -- but not a continuum. That's why many mental disabilities are the result of very concrete genetic variations. You can't have 80% of a generic variation. You either have it or you don't, which implies some sort of inherent discontinuity.

My guess would be algebraic ability is a normal distribution. But that's not a death sentence. People who might, because of genetics or more likely nurture/circumstance be unlikely to learn a lot in algebra, are not doomed to that fate. It just means it might require a lot more time and hard work for them and their teachers -- to the point where they might not find it feasible, and instead choose to do something else.

Your belief should probably be considered heretical, even among your conservative colleagues, because your colleagues should recognize how deficiencies in one area can be made up for in other areas, sometimes (but not always) at a higher efficiency cost.

Where genetic gifts are lacking, determination and perseverance can almost always make up the difference, especially at a basic algebra/calculus level.

I have been trying to get better at chess lately. I grew up playing it so I'm already fairly proficient, and I've been putting a decent amount of free time into studying openings and playing puzzles. Yesterday, I played my friend who hasn't played in literally years and never really played seriously anyway, and he beat me handily 3/3 games. I know that I do not have the brain to be a chess master, no matter how much time I put in, because I don't have enough working memory to keep the board in my head.

Life circumstance stratification, certainly, and you will get no disagreement from me that some people don't have the time or energy necessary to devote to something as not-immediately-useful as calculus, but from an intellect perspective, no.

Everyone who is not experiencing some kind of mental illness has the intellectual capacity to learn calculus, though I will capitulate it is fortunate for me and this statement that we do classify a low enough IQ as a mental illness!

Are you seriously suggesting trying to raise multiple children while simultaneously learning calculus is approximately as hard as living alone and trying to learn calculus?

Honestly, I think you're trying to find a neat solution where none exists. You haven't stumbled on anything here, you're more likely ignoring the real-life circumstances your students find themselves in, and your colleagues are not.

To suggest it? No. To be taken seriously though?

I could be living in a ditch down by the river, or I could be the leading researcher into how people learn calculus and it wouldn't make a difference with regard to my argument (though, practically speaking if I were the lead researcher on how people learn calculus, I'd be more likely to back my argument up with objective research, and that would strengthen my argument).

> (of an argument or reaction) directed against a person rather than the position they are maintaining.

Nothing about a reputation or using it to discredit them.

For example, I'm not a calculus teacher, therefore I couldn't possibly form a valid argument about how teaching works. That would be an ad hominem, because it focuses on who I am (not a teacher), rather than what I've said (the teacher I was replying to hasn't eliminated any variables at all before drawing their conclusion). (It's also nonsensical, considering how many other things people teach, and how small a percentage of all teachers calculus teaching ends up being, and how unrelated-to-the-science-of-education calculus is).

It's actually funny, because the example you've given isn't an ad hominem, since you have evidence to support the idea that I can't cook (the burnt toast). You're equating an absence of information about me with specific data, which is different.

People have to make judgments with imperfect knowledge. It’s reasonable to discount the unsubstantiated opinions of someone with no experience with the topic at hand.

It should at least be interesting to you that it was obvious from your comments that you don’t have experience teaching mathematics in the classroom. Why was that so obvious to those of us with that experience? The previous question is rhetorical.

What is interesting to me is the fact that you retreated to this ad hominem the moment you were challenged, because it tells me you don't have any real explanation for how you eliminated other possible causes for your students sometimes performing poorly.

You'd prefer to live in a world where your experience has meaning than to live in a world where your experience is not valuable when faced with this question, which is completely human of you, but ultimately not useful in this discussion, due to its anecdotal and un-rigorously collected nature.

I can't stop you from throwing this New Yorker article, and the other works of Dr. Harden, in the face of your colleagues, but I can hopefully dissuade others from making the same logical mistakes you're making. I believe I've succeeded at that, by clearly highlighting the carelessness of what you've said here.

Ultimately, what I find most fascinating is, in real time, you've demonstrated how right Dr. Turkheimer ultimately is and how dangerous this research can be when put in the hands of folks who don't understand its delicacy or even the basic facts surrounding these arguments.

I'm grateful for your engagement with me, it's been helpful to work through this with someone like you, but I'll commit to the thing you tried and failed to do; I'm no longer going to reply to your comments in this thread. You're clearly (and I mean clearly) wrapped up in a need to think of some of your students as too dumb to learn calculus, and there's literally nothing I or anyone else here can say that would convince you otherwise, and at this point I've done my part in preventing others from thinking that your insight is useful or helpful in this conversation.

Have a great rest of your evening!

Here’s an example:

… I must be a calculus teacher in order to challenge your assertion that IQ is the primary source of the problems your students have with learning in your classrooms..

No one has said any of these things and no one has implied any of these things. I never said or implied that IQ is the primary source of anything. No one believes that you must be a teacher of calculus to be right. What people have wondered is if you are a teacher because some of your statements seem to the the type of statements only a non teacher would make.

Given the immense complexity involved with learning, there's essentially no way for you to describe the situation with enough detail for anyone to be able to make an informed judgement. It's extremely common for bad players to describe themselves as 'proficient', spending a 'decent amount of time' can mean anything, studying openings and playing puzzles is likely the wrong way to practice, the skill level of your friend is impossible to ascertain, etc.

The way you actually get better at anything is by:

1. Dedicating enough time to it, consistently (~20h per week or so maybe if you want any sort of quick results). The incredibly common trap here is that people think that spending time on a game equals getting better at it. There's no way to improve without spending a certain amount of time playing, but spending time playing does not make you better on its own.

2. Doing whatever it is you want to get good at (doing anything else does not count - if you want to learn how to play regular chess, play regular chess; puzzles or anything else does not count).

3. Reviewing your games to find mistakes you've made - this part is crucial, if you can't see any mistakes you made, you cannot improve. If that's the case, get someone better than you to review your game(s).

4. Playing while trying to work on addressing one mistake at a time, until you don't make it anymore.

Chess is actually really easy to improve in - fast, trivially repeatable games, chess engines, lots of learning resources, objective rating system.

> I know that I do not have the brain to be a chess master

I'm not sure what exactly you mean by 'chess master', but for example getting into the top 10% of players is pretty easy. You are indeed incredibly unlikely to be one of the best, because for that you will have to dedicate your entire life to playing chess. But unless you literally want to be one of the best (top 1%+ of players), genetics will not limit you. You may take more time to get to any given level than someone gifted with more working memory or whatnot, but it's still doable.

> because I don't have enough working memory to keep the board in my head.

Because of humans' limited memory and processing power, we play games not by exhaustively analyzing but via building simplified models and heuristics so I don't see why you'd need to 'keep the board in your head'.

I don't think the onus is on me to prove you wrong, so much as the onus is on you to prove that your experience is scientifically rigorous and representative of more than just your personal experience.

And to be clear, I don't doubt that many people "can't" learn calculus in the same way I "can't" run a marathon; we don't have the desire, discipline, and free time necessary to do the work required. This does not speak in any way towards our capacities to do so, only our desire.

So I suspect the vast majority of your students didn't have the sufficient desire to learn a lot more than they were incapable, and I think it's a critical distinction, because you can change people's motivations, but you cannot change their capacity.

As stated in my original comment I’m quite liberal. In education we talk a lot about social conditions and their effects on education. It’s why teachers’ unions strongly support universal healthcare, school lunch programs, etc.

After 20 years on the job I’ve come to the conclusion that some can’t learn it. This isn’t controversial in some sense since we know retarded people can’t learn algebra. So what level of capability is necessary to learn it? I don’t know but I suspect it’s well above being mentally retarded.

I could be completely wrong but given your lack of experience in the topic my experience ought to at least make you pause a bit. In another comment you wrote about me:

… you're more likely ignoring the real-life circumstances your students find themselves in, and your colleagues are not.

Talk about making an assertion with no evidence! At least I waited 20 years before stating my absurd assertion.

Also not for nothing, but "retarded" hasn't been a medical term for some time now, with institutions such as the AMA and SSA both replacing the term with "intellectual disability". The only remaining reasons you'd use it are either because you are thus uninformed, or you're signaling something...

I’ll read whatever response you have but won’t comment further. I wish you well. Keep up the good fight!

I think you, and everyone who thinks like you, need to take a very hard look at what you've done to eliminate the environmental factors related to your viewpoint that some people aren't smart enough to learn something, because its extremely easy to trick yourself into thinking someone is incapable when in reality other factors abound.

As the article explains, if society refused to educate/feed/raise/nourish red-headed children, there would be a genetic correlation between red-headedness and intelligence.

How absolutely confident are you that you've accounted for every explanation besides the genetic one in determining why some people can't learn algebra?

My university would conditionally admit students who had a math score below a certain threshold on their ACT, I think it was 19 or something. Anyway, as part of their admittance criteria, they had to attend an after class lab for an additional hour an a half for a total of three hours per week dedicated to learning pre-algrebra. There were four modules where they would do some reading, work through some example problems through interactive software, and then have some homework to work through that was almost identical to the examples. Students would try and try and try and try to learn the material, and they would take days to work through the problems on their own (often with my guidance, giving pointers on how to think about the problems) to finish the module so they could take the quiz and pass the module. Near the end of the semester many students made appreciable progress, but for others the inability to retain and apply what they've spent so much time on results in tears, especially because they don't know if this requirement will keep them from being able to graduate.

Given infinite time, could these guys all have figured out pre-algrebra enough to pass? Maybe. But the amount of time it takes them to learn math concepts that are very easy for us means that it's entirely impractical to expect them to ever achieve proficiency in advanced mathematics.

This just does not look like a universally effective way of teaching to me, irrespective of the topic. It's hardly any wonder that some people fell through the cracks if they were unfamiliar with the subject in the first place. What about leveraging stuff that's actually been tried and tested, like the Khan Academy videos and their automated interactive, school-like environment?

Your VO2max, max heart rate, and other factors appear to be significantly determined by genetics, and will absolutely contribute to your capacity to run a marathon. If it takes you a year of hard training, but it takes me a few practice runs a few weeks before, it's not fair to say we required the same amount of desire, discipline, or free time to succeed. I would instead say we had a very different capacity to run a marathon.

Are you saying that similar things could not possibly be true for learning math? Or really anything else that humans do?

All healthy people can learn calculus and run marathons, with varying degrees of success and effort, due to genetics and environmental factors.

I also don't think "healthy" and (presumably) "not healthy" are useful categorizations. There are many people in that fuzzy in between area between "healthy" and "not healthy", for both physical and mental health.

Is it really worth so vigorously arguing the semantics of "some people are incapable of learning algebra" and "it requires an impractically large amount of effort for some people to learn algebra"?"

You can overcome difficulty, you cannot overcome (by definition) impossibility.

What you keep ignoring in what I'm saying however, is that I do think certain things are impossible for some people. I will never play in the NBA, for example, but that's a far cry from being "very good" at basketball.

Learning calculus and completing a marathon are not the point at which "healthy" (and yes, it's fuzzy, but precision is impossible on this topic) people are sometimes unable to do things. Winning a marathon and getting a Ph.D. in mathematics, I would acquiesce to your argument.

In other words, I think anyone can dabble in anything, but you do need an alignment of genetic and environmental circumstances to be in the top 1% of something. I could be argued into top 10%, but below that, it appears the data supports almost anyone being able to do almost anything, or at least well beyond whatever artificial lines we might draw to discourage people from achieving.

To turn it around: Do you have any evidence that your "anecdotal evidence" doesn't simply reflect your teaching skills?

This is a strawman. They never said 'gifted', they just said that some people have it and some don't.

> deficiencies in one area can be made up for in other areas

Not what we're talking about. GP is probably great at things that aren't chess. That doesn't mean they're good at chess.

> Where genetic gifts are lacking, determination and perseverance can almost always make up the difference

Determination and perseverance aren't (at least partially) genetic gifts?

Honestly I think 'genetics' are a bit of a red herring here anyway because the real underlying assertion being challenged is that all humans have equal potential in all things (which anyone who has ever taught any mildly difficult topic will know is trivially disprovable). The precise mechanism whereby innate human potential differs isn't important if you can't even agree that it differs.

My english teacher/class master till 8th grade insisted I follow a profession that doesn't involve math -- she even recommend me to be a radio host for a children program -- she thought highly of me, but she thought math was not my thing.

She was half right -- I have ADHD and even though my blood relatives are highly successful (doctors, lawyers, judges) NOBODY was good at math.

Anyway, my parents thought otherwise and promised to buy me a PC if I get admitted to a math/programming class in highschool and paid for math tutoring. My math skills completely turned around in a year and I ultimately loved math, especially calculus.

I don’t get to decide who has the ability to learn algebra. All I get to do in this regard is determine who knows enough to pass.

First, there are clearly a large number of people out there who have no business trying to pass algebra 101. Even if they theoretically could with sufficient effort and tutoring, it is not worth their time, nor is it going to lead to that much that is beneficial.

Second, teaching is already hard. Teaching math also requires an understanding of math that is rare. Hence finding people who are good math teachers is hard. Finding people who are good math teachers and willing to do it for a math teachers salary is even harder. If it were trivial to improve our teachers we would off course do it. But it is not at all trivial.

I think this matters less in a liberal arts college, but especially if you are going to ask students to build on their algebra 101 with other courses. If passing algebra 101 took a lot of effort, chances are that the other courses are going to take the same amount of effort, if not more.

Then you evaluate and attempt to improve techniques as is done every year.

I truly don’t believe my people are genetically smarter because we have a truly awful abysmal track history in governing.

My guess as an immigrant who went to community college here and then to University here is that students just have too much choices/freedom. They are told they should strive to whatever they want. I and my friends grew up told you need to choose a major that will bring bread on the table, that should be the only priority.

This is the "As a mom"-argument applied to iq. When it comes to child-rearing mothers will not infrequently claim to know what works best in general because they have had children of their own. I mean an equally valid interpretation of your anecdata is that not all people can learn how to teach mathematics at community colleges.

My collection of anecdotes that not everyone can learn math works as an anecdote that not everyone can teach math.

Of course there are genetic influences and not everybody can be an Einstein or rocket scientist. But below that level there is a lot possible that has nothing to do with genetics. Most things are more influenced by motivation than the ability. And on that the real cause does not matter.

If the people would spend only a fraction of the energy, they waste with questioning if something is genetic or not, in the development of skills by learning - they would improve beyond the proclaimed "genetic" level.

Reduce media consume. Reduce politics. Almost the whole discussion is toxic. The right proclaims everything is genetic but on the other side punish people not having the "good genetics".

It's worth noting however that we often consider these people "disabled" (or euphemistically "differently abled"). The point is that there's no genetic dial you can turn from "worse" to "better", it's more like a massive board of switches that feed into each other.

Like if I learn accounting, I know why I'm learning it; to learn how to create, audit, or read the financial statements of a business. If I learn math, it comes across more as "here are these arcane puzzles you need to solve." Like no one does double entry bookkeeping in the abstract; you learn the history, the methods, etc in the confines of an express purpose.

But trying to learn precalc as an adult (which i do get), what's striking is how little purpose or context there is to it. Why do I need to know how to factor imaginary numbers, or know the slope of something?

I think math when isolated is a big reason why its so hard to learn. They teach the toolset but people don't have the need to use the tools unless they go into a separate subject.

Why penalize people who are trying by failing them out of the simplest classes?

Rich idiots fail up, poor idiots end up homeless. No reason to penalize the poor who try when the entire game is arbitrary. This position tends to make certain groups bristle, especially those with a classist sense of "fairness".

It's so funny watching people scramble to avoid admitting that genetics has a huge impact on humans and their potentialities.

Granted, as a species, we are the closest to blank-slate out of any species ("niche-switching is our niche"), but reality doesn't go away just 'cause we don't like it.

A good deal of the folks enmeshed in various delusions related to their belief that reality is socially constructed, I've found, are folks that have little concrete experience with reality. Academic types, those who've exclusively worked in knowledge-production or in offices. Rock climbers and farmers are very much not prone to these delusions, for a couple of examples.

Try to convince a dog breeder that dopey English Mastiffs are just an environmental change away from gaining the intelligence of the German Short-Haired Pointer, which can practically solve Sudokus.

What about the anecdotes of millions of people who self-profess that despite very much effort, they just can't wrap their head around some advanced math concepts? That doesn't count as evidence?

I don't know how to weigh these anecdotes, but I think that's suggestive that the methods of teaching might be relevant even to people who struggle with math for decades.

If you were arguing that math comes easier for some people than others, sure that's strong evidence. If you're arguing that they are literally incapable, and no set of curcumstances would allow them to learn - that is a very different claim and needs very different evidence.

But I would speculate that of the set of people who bombed the class, they could fall into a number of buckets. E.g., one bucket is people who were mentally capable of learning the concepts, but were to lazy to put in the effort (then we can quibble about whether inherent laziness puts people into the "not capable" bucket). Another bucket is people for whom alternative learning environments might have brought them to understanding and a passing grade. Another bucket is people who just lacked the preliminary background and with a couple years of effort could be made to pass the class as it exists. And finally, another bucket is people who are genuinely incapable, regardless of environment, of understanding the concepts.

This shouldn't be surprising. I have tried to deeply understand quantum mechanics, and while I can parrot some of the most well-known and more simple concepts, I truly believe that I lack the capability of grasping the very core, deep insights in an intuitive way. I might pass undergraduate level classes in the topic, but I am fairly certain I couldn't achieve a PhD. I'm not the dumbest bulb in the shed, but I can see that there are people much, much brighter than I, and it is obvious that their ability to understand more advanced and deep concepts is greater than mine; This leads to the observation that of the set of understandable knowledge in the universe, some of it is available to some people but not available to me, no matter how hard I try. (I take solace in Feynman's quote, "If you think you understand quantum mechanics, you don't understand quantum mechanics.")

Then look at every human capability and its distribution across the universe of humans, and it's pretty clear that we all can't do everything, every one person has some cutoff beyond which they aren't capable of understanding in any given topic. For some people (and hey, maybe it's a really small slice of the population), that cutoff is somewhere before Calc II.

And we both admittedly don't know the size of the bucket of people who cannot, given a lifetime of 80 years continuous study and tutoring, understand a topic.

But in my teaching experience, it's dwarfed by the group of people who doesn't care about the topic and bombs because they don't put in the effort.

Students need to focus on the basics and no where is this more true than in mathematics. Too many students muddle through middle and high school mathematics without gaining mastery or with multiple gaps in their knowledge. By the time they get to calculus they're simply unprepared to put all their previous knowledge to use. "Calc 101" is the first time that many students are required to apply theorems and then use algebra, trigonometry and arithmetic to arrive at results to problems. If there's any weaknesses in their fundamentals it's going to make the problems intractable (and, yes, they are for many students).

It's better to track students and only advance them to the next level in math when they've demonstrated mastery of previous topics. That would mean, of course, that a good fraction of students would never "reach" calculus (or even algebra)-- but that's OK if it means they have enough numeracy to balance a ledger or learn avoid blowing money on the lotto. At least their time would not be wasted on trying to do "Business Calculus" in college.

Shows, also, the silliness of this argument. Some people can pick up calc at age 10 no sweat whatsoever. Others struggle mightily with the basics in their 30s. Should we as a society invest 1000x the resources in the strugglers to ensure they can achieve the same understanding?

There's no need for everybody to reach the same level of understanding, but I think the pandemic has shown the importance of teaching as many as possible the basic concepts of calculus. "Flatten the curve" doesn't mean much when you've never heard of integration. The same applies to climate change. People will have more faith in mathematical models if they think it's something they could have done themselves if they really wanted to (overly optimistic judgement or not) instead of some bullshit the so-called experts made up to bamboozle them.

By the time you’re in college, it’s already over. It’s why good daycare / preschool / school for everyone matters, otherwise you’re just missing out on a lot of potential talent.

You know, if you were trying to teach Chinese to 18-year-old English-speakers at community colleges with three hours a week of lecture, you might come to the conclusion that only a few rare geniuses had the ability to learn Chinese at all, and none would ever learn more than a few hundred words. But of course over 98% of people born and raised in China learn to speak Chinese fluently by the age of 5, and that's not because of genetics; it includes almost 98% of white people born and raised there too. There are several factors that I think of as key to this difference:

1. Plausibly there is a critical period for language acquisition, and if so, it very likely ends before age 18. Looking at an 18-year-old you can't tell whether their deficits in Chinese-speaking ability are due to genetics or environmental effects in the previous 18 years. There probably isn't an early-childhood critical period for calculus (I think you need formal operational reasoning for that) but maybe there are other things you need to develop early on for calculus to be easy for you when the time comes. Like, fluency in reading, for example.

2. Three hours a week, 36 weeks a year, for two years, is a total of 216 hours. Native language acquisition more typically involves over 20,000 hours of language exposure by age 5. Sometimes it's surprising how much more you can achieve when you apply literally over a hundred times more effort. Or, to look at it a different way, how little you will achieve when you're applying less than 1% of the effort necessary to get good results.

3. Schooling is a really terrible way to learn things. Extrinsic motivation displaces intrinsic motivation, massed practice displaces spaced practice because it gets better exam scores (especially if you know when midterms are coming up), the feedback necessary for improving is delayed by hours or days by the nature of homework grading, and the pacing is inevitably far too fast for some students and far too slow for others. It's well established that individual tutoring produces results two standard deviations better than ordinary classes (Bloom's two sigma problem). That's 30 IQ points. And Bloom wasn't even spacing out practice over decades the way you ought to; he was working within the semester structure of traditional schooling, extrinsic motivation and all.

Is there a possible human culture where 98% of everybody routinely learns to do symbolic integration? Or does human nature render that impossible? Maybe. (Hell, when I have an integral to do in my head, I myself invariably settle for just approximating it unless it's a fucking monomial. Maybe I should spend a few semesters doing them on the blackboard in front of a class, I bet that would help.) But the meat grinder of community college doesn't give us much evidence about that one way or the other, except to know that we don't live in that culture today. It doesn't help us at all with the question of what to attribute to environmental effects and what to attribute to genetics.

(One indication that such radical changes may be possible is the gradual transformation of literacy; hieroglyphics were the province of the priesthood and the quasi-priestly scribes, and even after the invention of alphabets, Charlemagne and Genghis Khan were illiterate. Can you imagine trying to teach a classroom full of 18-year-olds hieroglyphics in three hours a week, if they had no previous experience with reading and writing? Yet today literacy rates are over 95% in most countries, though countries with non-phonetic orthographies like Chinese and English lag a bit behind.)

Now, to be sure, some people and some books are better at teaching than others, but it typically has nothing to do with notation used, and everything to do with the order of introduction of concepts, level of detail of explanation (which can be both too high and too low), amount and quality of examples, etc. However, the core issue here is that some things are actually genuinely hard, and people of average intelligence simply cannot grasp them without expending ludicrous amounts of effort.

If you have some concrete suggestions about mathematical notation, ways it could be improved in more than superficial manner, I (and the rest of mathematical community) is very much open to hear them. Improvements in notation do happen regularly, and when they are valuable, they reach wide acceptance. For example, in the second half of 20th century, the notation of commutative diagrams have been invented, and it spread like a wildfire, because it genuinely facilitates understanding.

I think its pretty clear that momentum causes a lot of nomenclature pain. You can't just redesign entire fields of understanding every couple decades. For instance, what other fields use single greek characters to label concepts in a seemingly unpredictable arrangement? Often, local maxima are found because concepts are added in the context of the field as it already existed.

And its not just math. Most sciences have this problem. It is what it is but its silly to say we're in the best of all possible worlds just because math has been around a long time.

This is a perfect example of what I was talking about. Greek letters are typically used in opposition to Latin ones in order to distinguish what programmers can think of as “type” of concepts. For example, when doing geometry, you might want to designate angles with Greek letters, and points or line segments with Latin ones. This makes it much easier to mentally keep track of what name correspond to which object.

Yes, learning Greek letters for the first time is some amount of initial overhead (not much, as typically used ones are similar to Latin anyway, nobody starts with psi or xi). However, crucially, this overhead is paid once, and pales compared to the difficulty of learning the concepts being represented in the first place. It is never the case that replacing Greek letters with Latin makes students go “oh thank you, now I understand everything!”, instead, things are typically just as hard as before. However, replacing Latin with Greek might actually do that, by reducing the mental overhead through introduction of categories (types) of objects.

Hard disagree. The most important thing I learned is that it is all made up on the spot to the point that the lecture material explicitly says that books have used 6 different forms of notation for the exact same concept. When you understand that you drop any pretense of "design" in the notation. That helps you abandon foolish ideas that it is "consistent" and that the only thing it is optimized for is the author. When you understand that then it's just a meaningless barrier to overcome but it also becomes easy to overcome precisely because it is that trivial. You just get used to it and e.g. learn the alphabets of the dozen languages (including klingon because the lecturer had to make that joke) from which the variable names where sourced from. Once you did the meaningless grind the barriers are gone.

> People who complain about notation usually actually have problem with the substance, and the complaints about notation is just a coping mechanism, to deny hurtful reality that one can’t understand something hard, blaming external factors instead.

No it is quite simple. You can't understand an easy or hard concept if you can't read it. I still remember how I understood nothing in the first semester. Then when I was preparing for the exam everything was extremely easy because the notation was understood by that point.

>If you have some concrete suggestions about mathematical notation, ways it could be improved in more than superficial manner, I (and the rest of mathematical community) is very much open to hear them.

As I already said that is meaningless because there is no universal notation. "The mathematical community" will adopt a fraction of proposals and further splinter into separate "factions".

Because they are unable to change it. Just like with any thing evolved over a long time, like music notation, languages, even, to some extent, programming languages. Every change brings a lot of pushback, it's a monumental task to create a new one and even more so is getting any traction with it.

That doesn’t square well with the fact that notation keeps getting refined and improved. There is no pushback for genuine improvements. Biggest problem here is that there rarely are changes that clearly and meaningfully improve situation over status quo. I gave one example above, but overall, I am not going to take complaints about notation being obstacle to understanding seriously without concrete ways how to meaningfully improve it. You can of course keep complaining that it’s confusing, but without proposals for improvements, you’re actually complaining about the difficulty of substance, not the notation, and it says more about you than about notation.

Admittedly, that's from physics, so we can't really blame mathematicians for it.

Sometimes notation elides 'obvious details'. The details are obvious to those that have already gone through the learning curve, which is easier if one is strongly connected to other working mathematicians than for outsiders. Mathematicians barely notice inadequate or unusual notation. Outsiders struggle and spend significant energy just deciphering the notation.

Anecdote 1: Sometimes notation is too terse: single letters. Granted, efficient for whiteboard scribbling. Would be really nice to standardize an appendix for notation. E[X] = <expr>. Hmmm, what could E be? By the fifth paper, somebody bothers to write 'expected value' in plain English and the mystery in unambiguously clarified. In a voice-based interaction this is a non-issue, not so for those that only have text to deal with. This compounds as a novice has to juggle a set of mysterious symbols with tens of elements.

Anecdote 2: Long long time ago (before ubiquitous email or http:// took off) a young student spent some time working through a handful of type theory academic papers that somehow trickled into his corner of the Universe. The type inference rule notation (https://en.wikipedia.org/wiki/Type_rule), which is rather straightforward in retrospect, ate up more time that he's willing to admit. Γ is just a set of judgments and Γ |- expr is just a notation for 'Γ includes the judgement expr'? Then why don't they use the standard expr ∈ Γ, this is confusing? Are there some examples? Is there some code that one could possibly run through a debugger? The questions remained unanswered, as there was noone he knew in the same linguistic sphere interested in the topic. And the papers themselves never detailed such obvious notation details.

PS. I agree that the bigger obstacle is the lack of proper big picture 'why do we even bother with these concepts / theorems'. At an extreme, there is (used to be?) a certain style of math books consisting exclusively of a dry litany of 'Definition 1.2.3', 'Theorem 3.2.4', 'Corollary 2.3.1'. Very rigorous and very difficult to ascertain what problems they were trying to solve.

I really cannot conceive how one can learn what the concept of “expected value” of a “random variable” means, without encountering E[X] notation. This is a technical concept having a technical meaning, and any place that actually defines this meaning will teach you this notation. If you see this notation for the first time in some academic paper, but haven’t ever read any probability textbook, it means that you almost never actually learned the concept, which is my entire point. You might have some intuitive understanding derived purely from the literal meaning of the words “expected value”, but without actually getting technical, this intuitive understanding is mostly superficial, and, as such, not very useful. You won’t be, for example, be able to answer such fundamentally important questions like “is expected value of sum of random variables a sum of expected values of each? Is expected value of product a product of expectations?”. You can’t know answers to these questions without having ever seen E[X] notation, and if you don’t know the answers, your problem is with the actual concept, not notation.

> PS. I agree that the bigger obstacle is the lack of proper big picture 'why do we even bother with these concepts / theorems'. At an extreme, there is (used to be?) a certain style of math books consisting exclusively of a dry litany of 'Definition 1.2.3', 'Theorem 3.2.4', 'Corollary 2.3.1'. Very rigorous and very difficult to ascertain what problems they were trying to solve.

I agree that it very much often is a problem. It’s not a problem of notation, though.

Sorry, but I didn't learn E[X] notation. I learned E{X} notation. Notations aren't even consistent across the lectures I have attended.

"But that's trivial!" Not to students who aren't future HN readers.

That a well-informed position believes that not (almost) everyone who is attending a college can learn single-variable algebra is fairly depressing.

I don’t believe everyone has equal intellectual talent in all areas. My talent for math far exceeds my talent for physics. I’ve tried to learn physics but I just can’t. I have no intuition for it. There is variation amongst our brains and how the connections in our brains formed in childhood. In some people the wiring is such that learning a given topic is not feasible. Such is my belief.

I'm not disputing your superior experience on this topic (hence "well-informed position"); I'm just saying that it's fairly depressing.

I understand now what you meant by fairly depressing. Thank you for the clarification.

In general, you'll find that "leftists" believe that psychometrics is bogus in terms of science. Instead, it masquerades as science in order to fool the public into believing that biased public policy is neutral. There's no denial that some folks have brain damage, developmental disabilities, etc. but a denial that standardized tests are an appropriate proxy for genuine medical diagnoses.

Community colleges should present themselves as a public benefit. We shouldn't ask that everybody pass any class, simply that everybody has the opportunity to attend/audit any class. It is unfortunately true that the typical university administration is clueless and money-grubbing, preferring graduation rates to other metrics.

I further suspect but didn’t state this that if one doesn’t have exposure to algebra growing up then it is extremely difficult to learn as an adult. Children are easier to brain wash and in low level Mathematics we do a great deal of brain washing.

* IQ causes income

* Income causes IQ

* Some unknown thing causes both income and IQ

* The correlation is spurious; IQ and income are unrelated

Evidence strongly suggests that the third bullet point is true; socioeconomic class causes both income and IQ. Richer people living in nicer neighborhoods both have better opportunities for income, and also better opportunities for education; education causes IQ. This is why redlining is brought up so often as a root cause of so many of the disparities in quality of life; redlining deepened socioeconomic divides.

Adoption studies might also be informative.

Isn't the whole point that you can't simply look at the gene of a person and know what his mind is capable of except for genetic defects and even then it's hard to estimate.