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Notes -
On (Noticing) IQ (When You're Allowed To)
I was writing a reply to @TowardsPanna 's question in the SSQ thread that got a bit out of control. Large enough that I decided to just post it here too.
For what it's worth, none of these ideas are particularly new, at least they aren't if you compulsively browse LessWrong (or even read a lot of Scott), but the older I get, the more I realize that novelty is often just a nice-to-have.
Yes. Of course?
Humans aren't stupid. We’re expert pattern-matchers. We’re distinctively evolved to be relentless Bayesian updaters, constantly scanning our environment for correlations that offer a survival or status advantage. We’re Noticers™. The problem is that we’ve built a civilization where noticing the most predictive variable for life outcomes is considered a faux pas at best and a moral failing at worst. It’s common now to explicitly state, in corporate policy or legislation, that specific classes of Bayesian evidence are "fruit of the poisonous tree" and have to be ignored.
Like many well-intentioned interventions that hinge on obscuring reality, it doesn't work: when you outlaw the most accurate map, people don’t stop navigating. They just buy worse maps.
In the ancestral environment, we didn't have the Raven’s Progressive Matrices (only the Modern Hunter Gatherer has had the privilege). We did, however, have proxies. We had vocabulary size, wit, speed of learning. In the modern environment, we’ve muddied the waters. We’ve got proxies that look like intelligence but are actually just measuring conformity to the upper-middle-class Blue Tribe aesthetic. We look at credentials, which are increasingly just a measure of "willingness to sit still and follow instructions for four years."
(Please note that I don't think that education is purely credentialism. This is a disease that affects midwits first and foremost. A bachelor's in physics is a stronger signal than one in sociology.)
So when we agree to pretend that "mental horsepower" is a myth, or that it’s entirely malleable via the Growth Mindset, we don’t actually create a flat hierarchy. We just create an opaque one. We create a hierarchy based on the performance of competence rather than the reality of capacity.
This leads to a specific kind of societal gaslighting. The narrative is that anyone can be a doctor or a quantum physicist if they just work hard and acquire knowledge. If someone fails to climb the ladder, the implication is that they stupidly chose not to grow their brain. They lacked grit. They lacked character. In a world where aptitude is taboo, failure’s always a moral deficit.
The obvious analogy is a world where the Paralympics and the "normal" Olympics are wrapped up into a single event. Anyone, including the disabled athletes, who points out that missing a leg makes one very unlikely to win in the 100m sprint is immediately walloped and then piled-upon, with cries of "the Science! Not-Murray et al clearly showed, in their landmark 1998 paper, that the number of remaining lower limbs had no relation to performance in the sleighing, shooting and arm wrestling categories. The 2012 Olympics were studied in great detail, and it was confirmed that Usain Bolt had the same number and arrangement of legs as his closest competitors."
When the person with no legs fails to win the 100m sprint, the commentators nod gravely and say, "He just didn't want it enough. He failed to manifest the victory."
I see three main mechanisms driving this, and why the average person (even the smart average person) underestimates the sheer weight of g.
The "All Men Are Created Equal" Overcorrection:
We’ve engaged in a concerted propaganda campaign for decades suggesting that because moral equality is mandatory, biological identity must also be mandatory. This makes it practically unthinkable to consider that it’s perfectly possible to construct a political system that grants equal rights to all citizens while acknowledging that not everyone can visualize a rotating 3D cube with equal fidelity. But biting that bullet feels dangerous to many. It feels like opening the door to old aristocracies. So instead, we pretend the differences don't exist, which inadvertently creates a new aristocracy of "people who know the right shibboleths."
The Bubble and the Range Restriction:
This is the big one. We live in intense cognitive bubbles.
If you’re reading this, you probably spend your life surrounded by people within one standard deviation of your own IQ. You went to a selective university. You work in a cognitive field. You live in a zip code priced for high-earners.
You can go days, maybe weeks, without having a meaningful conversation with someone significantly below an IQ of 100, or even +- 1SD outside of yours.
This creates a statistical illusion known as range restriction. If you look at the NBA, height doesn't correlate very strongly with points scored. Everyone’s already tall; therefore, the variance in performance seems to come from practice, grit, and shooting form. If you looked at the general population, height would be the single biggest predictor of basketball ability (because the 5’5” guy isn't getting the rebound).
Imagine if the NBA had a rule that you couldn't mention height. You'd still need to pick players, so you'd start using proxies. You'd talk about "reach" and "wingspan" and "vertical leap." You'd notice that certain players had better "court vision" or "positional awareness." And all of these would be real skills, but they'd also be correlated with height, and height would still be determining who made it into the league in the first place. The difference is that now you'd be pretending you weren't noticing height at all, which would make you more likely to mistake the packaging for the product.
In your life as a student or a professional, you look around at your peers. You see that Alice is doing better than Bob. Since Alice and Bob are both roughly equally smart (they both made it this far), you attribute Alice’s success to her conscientiousness, her charm, or her work ethic. You conclude, erroneously, that "intelligence doesn't matter much, it’s all about hard work."
You don’t see the people who were both dumber and less hardworking than you; they’re in a different social class entirely. You rarely see the people who are simultaneously smarter and more hardworking; they’re running hedge funds or solving alignment theory and don’t hang out at the hospital cafeteria. These are far more likely to be acquaintances rather than peers.
The problem is when we generalize from this filtered view. We start believing that because IQ doesn't predict success within our bubble, it must not predict success period. We see a colleague who's a bit slower but works incredibly hard and does fine, and we extrapolate that to everyone. We forget that the slow colleague is still in the 85th percentile of the general population, and that the person in the 30th percentile who works just as hard isn't a doctor at all, they're doing something else, probably something that pays worse and doesn't sound as impressive on a Hinge profile (see proxies re-emerging?).
The Opacity of Alien Competence:
Some professions are more segregated than others. An ER physician arguably sees a more representative slice of humanity than a family medicine specialist, who sees a more representative slice than a Google L10, but the direction of judgment is difficult.
When a patient comes to see me, I’m performing a distinct ritual. I listen to symptoms, I peer at them significantly, I type things into a computer. To the patient, the output seems to appear from thin air. A sufficiently competent doctor makes it look like they never broke a sweat. A less experienced one (like me) makes up for it by projecting confidence and then panic-Googling the interaction effects of MAOIs in the bathroom later.
The patient can't judge my raw processing power. They can only judge my bedside manner, my clothes, and my degree on the wall. They’re judging the paint job.
When I'm on call and a patient is rambling in a disorganized way, using neologisms, showing thought disorder, I don't need to know their IQ score to recognize that something is cognitively wrong. But when I'm in the doctors' lounge, listening to two consultants argue about whether a patient's depressive symptoms are primarily biological or reactive, I also don't need IQ scores to know who's making the sharper arguments. The difference is that in the first case, we're allowed to talk about cognitive impairment. In the second case, we're allowed to talk about "clinical judgment" or "experience" or even "medical education," but we're not supposed to talk about the underlying mental horsepower that makes one doctor better at diagnostic reasoning than another.
(Ideally, the whole point of the medical education system and associated exams is to put the gatekeeping before patient contact. When you see an MD in the wild, you ideally want to not need to bother with asking for a transcript of their grades, nor fear that they don't know the safe limit for paracetamol. The reasons why this idyllic state of affairs doesn't hold are too lengthy to fit on this margin. Be smart about things.)
Conversely, I can't judge the competence of a senior software engineer. I know enough Python to automate a spreadsheet, but can I distinguish between a decently competent programmer and a 10x developer without referencing their credentials or status? Probably not.
This brings us back to the original shower thought. If we can't discuss the engine, we obsess over the trim.
In a "blue" environment where social class is ostensibly "over" and never discussed, it’s actually the primary sorting algorithm. We use heuristics like "do they speak with a Received Pronunciation or General American accent?" or "do they know the current correct terminology for this specific social justice concept?" to decide who’s smart.
This favors the people living and breathing inside the dominant culture. It favors the legacy admission who knows how to dress and speak. It subtly closes out the dissenting voice from the outsider who might have raw supercomputer-level processing power but speaks with a regional accent, wears ill-fitting clothes, and hasn't learned the subtle dance of feigning humility while signaling status.
Some professions, like programming, are comparatively more sane/honest. You can have a perfectly decent career in FAANG if you don't shower regularly and speak with a lisp, assuming you are actually good at your job. Hell, like the dude with the MLP (pony, not perceptron) resume, you can counter-signal by being incredibly eccentric. If you're still drawing a seven figure salary, then you're worth it.
Speaking very generally, I think society (a vague term, can't help it) needs to move in the direction of more meritocracy, not less. In practice, that looks like more standardized testing, with reduced focus on vibes.
The vibes are, paradoxically, easier to fake than the exam results:
Consider the standard critique of the SAT: "It just measures how rich your parents are."
This is statistically false. The correlation between SAT scores and parental income is real but moderate (around r = 0.4). But compare this to the correlation between parental income and "being captain of the fencing team" or "having a polished personal essay about your volunteer work in Peru" or "speaking with the correct sociolect during an interview."
The SAT is a partially gameable metric. "Holistic" traits are entirely gameable products.
If you are a rich parent with a dim child, you cannot tutor them into a 1600 SAT. You can maybe get them from a 1000 to a 1150. But you can buy them a spot in a prestigious internship. You can hire a consultant to write their essays. You can ensure they have the "right" hobbies.
Since this is my soapbox, and you're presumably still reading this far, I also want to speak out against another approach towards modern fetishization of pseudo-meritocracy, or more accurately, aristocracy.
I should probably elaborate on the @2rafa position, since she's the strongest advocate (only advocate?) She's old money. My impression being not "my parents paid for Stanford" money, but "my great-grandfather's trust fund paid for Stanford's new rowing pavilion" money. She'll happily tell you that the problem with modern society isn't that we sort by intelligence, but that we pretend to sort by intelligence while actually sorting by a particularly vulgar form of striving that she finds aesthetically repulsive.
Her critique goes something like this: The modern PMC striver is a grotesque creature. The Tiger Mom. The résumé-padding, LinkedIn-posting, "crushing it" bugman who measures his self-worth in LeetCode problems solved and whether he made partner by 35. This, she argues, is what you get when you tell everyone that anyone can be elite if they just grind hard enough. You don't get equality. You get a soulless arms race of performative busyness, a cargo cult where the striver apes the surface of elite competence without acquiring the substance.
Better, she says, to return to a system where everyone knows their place. Where the sorting happens early, quietly, and permanently. Where you don't try to be elite, you simply are. Where excellence is demonstrated through effortless grace, not desperate optimization. The boarding school accent. The understated wardrobe that costs more than a car. The ability to discuss Proust and quarterbacks with equal nonchalance. The aristocratic assumption that if you have to ask, you're not one of us, and that's fine. It's a system that rewards being rather than becoming.
From her vantage, this is obviously superior. And I get the appeal. If you're already at the top of the heap by accident of birth, a system that freezes the heap in place is very comfortable. You don't have to worry about some brilliant kid from a refugee camp out-hustling your mediocre son for the last spot at Harvard Medical School. Your son's spot is secure, not because he's particularly gifted, but because he's yours. The system can quietly acknowledge his inherited position without anyone having to say the quiet part out loud. The "genteel" facade is the point, it transforms raw inherited advantage into a question of taste.
This is far from the worst approach. The elites are elites for a reason. That reason is often hereditary. Even total regime change and societal upheaval usually has the dispossessed children of nobility (or the elites) almost inevitably percolate back to the top. This is evident in data from places as far-flung as China, where the grandsons of pre-revolution landlords often outperform their peers. The previous status-quo was bearable, in some ways superior.
However, the aristocratic alternative often ends up parasitic on the very meritocratic machinery it despises. You still need high-g selection somewhere. You still need the engineers, the surgeons, the generals who can think. You can dress that up in tradition and patronage, you can recruit them as client talent, you can offer them a place in the court, but you cannot run a technical civilization on inherited gentility alone. At some point, reality reasserts itself, usually via catastrophe.
But here's where the rubber meets the road: I am that brilliant kid's son. My dad did out-hustle the mediocre sons of privilege. He clawed his way out of a refugee camp because somewhere, a grinding, impersonal system looked at his test scores and said: "This one. This one is worth plucking from the mass and polishing." He wasn't sorted by vibes. He was sorted by a standardized exam that didn't care about his accent, his hand-me-down clothes, or whether he knew which fork to use at a state dinner. It cared whether he could solve the problems in front of him, quickly and correctly.
The "genteel sorting" system that @2rafa prefers would have written him off before he started. He didn't have the right pedigree, the right consonants at the end of his name, the right summer internships. He had the wrong everything except the one thing that actually matters for medicine: the ability to hold a thousand variables in his head while making a decision under pressure. The exam caught that. The "holistic" process would have missed it, distracted by his lack of polish.
So yes, I have a personal bias. I believe in meritocracy because meritocracy is the only reason I'm here, writing this, instead of hauling crates in a warehouse or pulling weeds out of a farm. But my bias aligns with a principle: if we're going to have sorting, and we are, because complex societies require it, then let the sorting be honest. Let it be based on the thing that actually predicts performance, not the cultural markers that predict comfort for the existing elite.
The aristocratic approach pretends it's avoiding Goodhart's Law by refusing to articulate its metrics. But the metric is still there: it's called lineage. It's just a metric that can't be improved upon, only inherited. And while it's true that modern meritocracy is imperfectly gamed (that's what this entire post is about) the solution isn't to replace an imperfect but theoretically climbable ladder with a walled garden whose keys are handed out at birth.
The modern PMC striver is indeed a pathetic figure in many ways. But he's pathetic because he's been lied to. He's been told that credentials are everything, then handed a system where credentials are increasingly just proxies for the ability to acquire credentials. He's been told that growth mindset will make him a doctor, then sorted by an IQ test disguised as the MCAT. His sin isn't striving. His sin is believing the official story, and optimizing for the proxies rather than the underlying reality.
@2rafa's genteel system doesn't fix this. It just makes the proxies even more opaque and even more heritable. It replaces the MCAT with the recommendation letter from your Exeter headmaster. It replaces the LeetCode grind with the unspoken assumption that of course you'll summer at the firm your father's college roommate runs. It removes the last remaining pressure points where someone like my father could punch through.
What we need isn't a return to aristocracy. It's a return to honesty. Acknowledge that g exists, that it matters, and that it's largely heritable. Then build a system that finds the people who have it, wherever they are, however they present. Make the exam harder to game, not easier. Make the credentials less important, not more. And stop pretending that the alternative to vulgar striving is egalitarianism. It's very much not. The alternative is feudalism with better manners.
Sadly, none of this particularly matters in the long-run. The AI will "meritocratically" take your job, and will eventually do it better than you can imagine. My kids aren't going to college. Yours probably won't either. I find that reassuring, in some ways, short timelines taken seriously make a lot of squabbling moot. You can stop running so damn hard, the winner has a jetpack. Isn't that oh, so reassuring?
There’s nothing I can add other than to reiterate @Corvos and @orthoxerox’s replies. Because classical systems still involve effective, consistent multigenerational meritocracy (ie social mobility), they effectively offer almost the same competence with a huge reduction in unnecessary make-work. Kids studying for 5 hours a day after school is worse for society than them spending that time digging ditches and filling them in; the latter at least involves exercise. Yes, you are impatient, but that is more about you than it is about society.
As for me, I have one correction to make. I’m not old money! I’ve said this before, but I was born upper middle class; my parents became truly rich only in my teens. My mother’s family were a mix of middle class for many decades. My father’s family were a mix of shtetl dwellers and Italian Jews who mostly arrived between the 1820s (early for Ashkenazim) and 1890s (with a few later outliers), and who went on to make and lose several fortunes, but who were at the time of his birth as middle class as you can get (think a small town accountant or government worker).
The other thing I want to point out- and I see myself as more expanding on your point than making a different one- is that in developed countries, thé effective floor for functioning people is just not bad enough to complain about. The people who manually harvest crops for twelve hours a day for subsistence wages are foreigners whose children will work in a warehouse, and their children will work for the utility company, and their children will work in offices. Everyone except the fruit picker is doing fine. I use this example not to make a point about immigration, but because there simply are not large numbers of native poor who are functional enough to run this cycle.
@self_made_human does not come from this society. Rapid upward mobility whatever the cost makes a lot more sense if your country has a large class of subsistence farmers. We do not live in a country where working in a sweatshop is aspirational, we live in a country where a mediocre suburban middle class life only requires that one be… mediocre. This is why I’m so hard on ‘accommodations’ in academia, or on the oriental grindset. Your kid having less than perfect grades will not sentence them to decades of subsistence level poverty. It’s ok to be bad at school because the only thing you need to do to have an OK life in America is to… not fuck up. Literally, look up the research on the success sequence.
This is so well said. I should print it out and put it in my office for students to read.
I wonder how much of the culture war can be charitably phrased as: On the one side we have people striving to try to make themselves/America better and on the other side people worrying that this striving is going to fuck things up. (It's not too hard to find examples that work with both red/blue as either side.)
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Hi, can you expand on this point? My position is that there's obviously marginal utility in increasing/decreasing schooling duration/homework, but then that's a very individual thing where some need more and some need less. This individuality expresses itself in extra tutoring (some people need more time than the average to get something) or special classes (some people need less time than the average to get something and is ready for the next thing). Since society most often rule in averages*, then yes an arbitrary fiat of extra 5 hours system-wide is probably going to be bad, probably just as bad if kids spend 5 hours digging. I suppose what I am trying to get at is do you have a different more clarifying example of your position?
* It's been said before but the dream for education is obviously personalized individual study plan suitable for the person. AI and tech seems to be 1 or 2 years from being able to offer this.
First homework is stupid. Even worse it is pointless. It takes the most valuable time of your life for some absolutely marginal advancement in academic performance. If someone can't grok quadratic equations - solving 200 more at home will help? Making them suffer trough war and peace or crime and punishment will make them appreciate literature? That somehow staring at the physics handbook will make them understand relativity?
We treat knowledge as goal not as a tool. Tell the little ones that at the end of the course they will know how to make total synthesis of cocaine and you will have the most attentive chemistry class in the history of the world. And they will do their homework without even being assigned.
Most work you perform during your working years will be stupid and pointless, but you still need to do it. Better that children are taught early to swallow their pride and get shit done even when they don't see the point in it.
You might as well put blinders on the kids to prepare them for the fact many of them will develop vision problems as they get older, and force them to wear fatsuits to get them ready for the obesity many of them will settle into as their metabolism slows, and steal their lunch money to train them for taxes, and...
"Adult life sucks, so we should make life for children suck to prepare them for it" is such an insanely negative-sum, anti-child view that I am filled with shock and outrage every time it comes up. It's like you don't remember what it was like to be a kid, because you don't treat children like people.
From "Book Review: The Cult Of Smart" by Scott Alexander:
And from "Chattel Childhood: and the way we treat children as property" by Aella:
For what it's worth, having seen homeschoolers extensively- the most impressive specimens homeschool through early highschool, but if male they switch to some sort of hybrid system after that(community college/homeschool hybrid is the best). Girls are approval motivated enough to keep themselves on track without a formal class structure. Boys are not. Not self motivated- approval motivated.
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We really need to stop talking about education like we're just educating ancient nobles' children.
Or as if the fact that a handful of related universities in one single country were based on an idea two hundred years ago somehow makes that idea The One True Fundamental Truth globally today.
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Going by the digging ditches comparison, it's also pointless and stupid, but for many kids it would be genuinely more fun and engaging. That's because it has the possibility of a positive reward signal: finishing the ditch. But the quadratic formula is something many genuinely are just not capable of: there is no final "get shit done" point for it.
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And better still that we work on removing the stupid and pointless work as much as possible.
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This sounds like a hack Michelle Pfeiffer would try to educate inner city children.
What would actually happen is that the kids would think you're cool but the minute they had to apply any of the things you said in a more boring, sterile environment (and of course they must) a bunch would get tired and move on to something significantly more stimulating. I don't know what schools you went to but plenty of kids are fucking lazy and need the tripwire of mandatory homework. Plenty of kids would be hooked on every word in one class and then not care about another.
You may have been particularly intelligent and conscientious, most aren't. Unless we're talking purely about schools that cut those people out, kids need to be forced to push through their boredom because work will be boring.
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@quiet_NaN this is interesting to me because I feel like high-school level homework (and to some extent college level) was actually helpful for me.
Doing the readings in history and typing up notes really helped me remember history better Doing problem sets in Calc I-III and professional exams helped me remember concepts better than I would've otherwise (though I still forgot most of them within a few years) Doing coding assignments in CS 101 really helped me be able to code.
Maybe it's more about elementary and middle school homework?
I absolutely agree. I had a math class at uni and the prof gave us hundreds of problems to solve with a promise, that one of them will be exact copy on the exam and that we are allowed to bring solved problems to it. This actually made me go through all the problems. Copying the one I solved from notes on exam helped, but I had perfect score on all the other problems as well. Solving them for a week or so definitely helped. I caught several deficiencies, I taught myself more efficient ways to do checks mid-problem, I even consulted theory as problem solving brought understanding that went beside me at the time, and I just rechecked the theory because it was cool to have more thorough understanding.
It was kind of grueling, but I did not regret it. It was probably better than yet another Netflix binge session.
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There is homework and then there is homework. On one end of the spectrum, you have problem sets designed to help you master algorithms. On the other end, you have things like projects, which Scott correctly defines as "take this subject you already understand, a few sheets of construction paper, scissors, and a computer program such as PowerPoint, and combine them in whatever random way you want as long as it takes a minimum of six hours of time". And there are plenty of those in high school.
In fact, Robin Hanson has a post about how only math homework helps. It probably generalizes to other math-like subjects, like physics and compsci. But it does not justify the three hours of homework a day that kids receive from all their subjects combined.
I've seen the justification that can be paraphrased as "most of them will be doing uninteresting and uninspiring work 9-5, uninteresting and uninspiring homework helps prepare them for the fact that not everything is gonna be exciting edifying self-directed projects all the time".
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Agreed. Doing the individual work is when learning happens. Sitting in lectures has its benefits but we probably do too much of it in high school. Cutting classroom instruction time by 66% with more time to do the work during the day would increase learning.
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I think that math class is probably the worst offender for pointless busywork.
Case in point: Polynomial long division. 100% "we make you learn an algorithm as a proxy for intelligence", 0% "something you will need to know as a prerequisite for understanding something else." The correct place for it would be "Having just discussed the properties of polynomial rings in general, here is as a curiosity a technique of dividing polynomials. You know that it will not be relevant for you exam because it involves just playing an algorithm (possibly even with concrete quantities)."
Instead you get tasks like "one of the factors of x^3+5x^2+7x-3 is (x+3). Factorize x".
People whose skill is to pass 'math' class in high school do not need to worry about being replaced by LLMs, because they were presumably replaced by WolframAlpha in 2011.
This is equivalent of doing multiplication and eventually memorizing the multiplication table during elementary school. With enough "busywork" you will be able to factorize from your head, which will enable you to solve some key problems much, much more quickly "look and see" style ,sifting through unproductive approaches before even trying them. This is cookie cutter thing in many engineering areas, basically anything that uses differential equations which is a really a lot of stuff.
Busywork is really important especially in early stages. You have to go through some shit like memorizing vocabulary when you are learning a new language, before you will be able to do some creative things like translating poetry or doing some Tolkien-like stuff with that language.
No, that is my whole point. Knowing the multiplication table up to 9*9 is a prerequisite for the following:
Anyone who is arguing that kids do not need to learn what 7*8 is denying the need for learning arithmetic at all. (There are probably people arguing that you do not need to know arithmetic because computers exist. I disagree. Knowing how to arrive at a result without a computer in principle, even if a million times slower and error prone, is useful in itself. Besides, with LLMs, this argument generalizes against anything you learn in high school.)
I am fine with busywork which is an instrumental goal for something worthier. I learned my multiplication table because I knew that it would enable me to do long multiplication in base 10. Then in fifth grade, they tried to make me learn 2..9 * 11..19, and I was not having it. If you are in base 10, the utility of knowing non-trivial products of integers takes a sharp dive after 99. I felt that if I gave in and learned what 137 was, they would come around in sixth grade and demand I learned by rot what 1315 or 237 was. (For factorization, the useful thing would be to quickly identify a prime factor in any integer in a given range, but even this obviously does not scale.)
A lot of the stuff kids learn in math class, painful as it may be, is actually useful in some fields. I don't think that youths should be forced to learn what a logarithm is, but they should be clearly told that any subject up to the softest social sciences will expect them to know, so that they can make an informed decision.
Other stuff has value because it teaches core concepts of math, such as proof by induction or the definition of real numbers. Sadly, people only learn what real numbers are when they go to university, and proof by induction in high school is turned into a mockery by turning it to just another pointless algorithm which can be used to prove increasingly pointless sum formulas by rote.
Wait - you value this, but not polynomial division? They're both things you can just ask the computer to do for you instead, but at least polynomial division requires you to hunt down a computer algebra system; long division capability come pre-installed on every phone.
Polynomial division is IMHO a curiosity.
Long division crops up every time you need want to split the bill.
Polynomial division might crop up in the wild if you want partial fraction decomposition, which I guess you might want if you are dealing with rational functions and want to numerically evaluate them or calculate their anti-derivatives. While I am sure that rational functions have their uses, my gut feeling is that they are both too narrow to pop up in physics a lot (where you will frequently have square roots placed so that your functions can not easily be transformed into rational functions) and too inconvenient to be preferred for empirical models.
Factorization of an integer is a hard but finite problem. Factorization of a polynomial is in general just not possible exactly. You can test if 7, 13, 17, 19, 23, ... etc happen to divide 3071 to factorize it. You can not test if x-1, x-1.1, x-sqrt(42+sqrt(42)) etc are factors of 5x5+4x4+7x**2-2x-2, because there are countably many algebraic numbers which could be a root.
I think that we learned both polynomial division and solving quadratic equations around grade eight. Solving quadratics in something which I would call bloody useful. Quadratic functions are the first non-trivial functions students can tackle, and quadratic equations pop up all the time in high school physics.
I strongly disagree with the sentiment that math skills which are less readily automated are more valuable. To grok (I'm reclaiming that word) how multiplication and division work doing long multiplication and division is definitely more useful than just using a calculator. Nobody needs the numeracy to be actually excellent at these operations any more. Anyone whose job actually requires them to multiply five-digit numbers will hopefully have the good sense not to try that by hand.
My more general point might be that I do not want students to be excellent at applying any algorithm. They will always suck very hard compared to the simplest of computers. Still, it is useful to demonstrate that you can apply an algorithm, even if it is just at toy-sized problems.
Also, applying a pre-learned algorithm is not math. Some algorithms (e.g. solving an equation for a variable) are genuinely useful in proofs, and thus are valid technical skills to learn to be able to engage in math, same as being able to write symbols with a pencil. And of course, 'can you apply an algorithm halfway reliably?' is also a good way to check if someone has a basic understanding of the algorithm in question (even if it does not probe if they understand why it works), which is why rotating trees by pencil is a staple in CS exams.
Still, for school math, I feel that 'can apply pre-learned algorithms' should earn a passing grade, not an actually good grade, which should require thinking.
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Where do you have proof by induction in high school?
IB does proof by induction in high school.
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In Finland in the mid 90s.
Of course that was by far the most useless thing they ever taught in high school math here (and in fact one of the few useless things).
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Any East Asian education system would touch upon proof by induction, and most certainly of any Math-specialized classes. Seems like AP Calculus briefly touches upon it. IB Math Higher Level and Further Level certainly does. The British A-Level Further Maths also does. India probably does.
Obviously teaching material quality varies and I'm not a math expert but proof by induction might not be accessible but definitely within reach for most high schoolers out there.
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You were probably member of temporarily impoverished upper class. I have seen many such examples as my country transitioned from socialism to capitalism after 1989. Optically your family may have been lower class as your peers in school during socialism, but your father/grandpa was a former math professor or successful entrepreneur who's whole property was expropriated by commies, and who was forced to become janitor or stoker/boilerman (a popular punishment by the party for people with wrong pedigree for some reason). You still had access to better homeschooling style education, you probably read books since you were 4 or 5, and you were bored at school thanks to access to huge library, you probably know how to play an instrument or two, and you know how to speak multiple languages - and all that despite barely having enough to eat. Nevertheless, you are in fact genetically upper middle class, that is your potential.
I think that especially people in US have distorted view of what true class means with huge number of immigrants, who often flee political persecution completely broke, and who are such temporarily impoverished upper class people. It is different from permanently poor chav/white trash lower class of people who are there for centuries, often due to their genetic issues. And I mean it literally - some of the most fucked up populations suffer from centuries of first-cousin inbreeding with huge accumulated genetic load.
In Paul Fussell's nomenclature, the class of a math professor is definitely the 'upper-middle' class, not the upper class. Math is much too useful to appeal to the upper class (except perhaps as an eccentricity signal 'I am feeling so confident in my place in society that I can pick a subject used by people who need to think to earn a living, because nobody could possibly mistake me for them').
They may not be as rich, but they were genetically up there with doctors and lawyers or upper management and other elite hired white collar professionals who are considered as lower-to-middle upper class. That is my whole point. I'd say it is culturally upper class similarly to composers or musical virtuosos and other gentlemen who exhibited enough brilliance that upper-upper class wanted to be around them and even marry them. There is similar story with Henri Pointcare for math - he was born into low upper class or maybe upper middle class according to your evaluation - with his mother being daughter of minor industrialist and father being professor of Medicine. Throughout his life Pointcare was rubbing elbows with the most influential politicians and elites. Niels Bohr had almost the same situation - father was professor at university and mother was daughter of Jewish banker. Bohr was also highly influential and definitely upper class when it comes to his contacts and power.
But in the end it is besides the point, I think you got the gist of the argument. There is a difference between rich family impoverished by revolution or relocated by Hitler, Stalin or some other pogrom, or even stricken hard by string of bad luck such as health issue, gambling or other addiction - and a true low class of literal inbred morons.
Doctors used to be and top lawyers are OWNERS. Math professors are not. That's why math professors aren't upper class.
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It is NOW, but back in G. H. Hardy's day...
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upper class? hahahahahahaha
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Makes me think of Hasan Piker, who appears to think that he grew up basically working class because his family was briefly of such diminished means that he was reduced to hiring the services of a second rate riding instructor.
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It's additionally confused when the immigrant/native thing gets treated as the only marker of class and/or is seized upon by first/2nd gens as a tool of giving them greater privilege. The amount of times I've had interactions with people talking about bias and classism in the West when they're top 5% in their country of origin and have been the beneficiary and performer of 1000-fold the oppressive activity on their own Untermensch.
Can we create a name for this?
The Pedro?
As in Pedro Pascal.
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Many such cases. You reminded me of the Claudine Gay - the recently axed first black woman Harvard president. Apparently her family were magnates from Haiti who owned a concrete plant in Haiti, with various shenanigans especially with regards to reconstruction after earthquake with Haitian politicians having shares in the company for some reason.
But yeah, she is marginalized person who needs affirmative action.
My law school class was full of black students who recieved explicit LSAT score adjustments in admissions and a frankly obscene amount of scholarship money/free resources by both the university and the law firms trying to pump up their diversity numbers. Not a single one came from an "underprivileged" background. They were the children of bankers and lawyers. One was a literal European aristocrat on their mother's side. It's a really radicalizing experience to be lectured to about oppression by someone who went to boarding school when you're the one two weeks behind because your student loan check hasn't come in yet and you can't buy books.
When I think back to what was the first step for me off the progressive reservation (this was back in 2013 or so), a coworker of mine was arguing that I (a man) was privileged compared to her. This is a woman who went to boarding school at Andover, spent most of her high school and college years snorting coke, and had parents who bought her a condo in SF as a graduation present. While I literally grew up on food stamps.
Amusingly, I’ve spoken to Latino H1B’s who were offended at the implication of being underprivileged.
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Many such cases at Google. It's absolutely bizarre for a public school/public college alumnus seeing those prep school->Ivy League women argue that anyone else is privileged. What's that you're saying? I can't hear you around that silver spoon. Oh, it's platinum, my mistake.
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I think that a case could be made that what made SJ so appealing to the upper-middle (and middle) class was that by denying class, it made them free to be as classist. Before SJP, expressing contempt of the working class would have gone badly in mostly left-leaning academic circles.
Now, it is totally permissible, because being working class does not make you oppressed. After all, with blank-slatism, everyone who does not manage to earn a degree in grievance studies despite not belonging to an oppressed group is clearly just lazy.
So if you are a white woman whose daddy paid for your college so you could work in HR, you a free to feel vastly superior to the proletariat in a way which would have utterly embarrassed any upper-middle person in the 50s. Just take care to refer to them as "deplorables" or "MAGA base" who can not even bother to educate themselves about what is really the secret handshakes of your class membership (like the updates to the pride flag, pronouns, or knowing when you are supposed to do a land acknowledgement) but what you conveniently pretend is the basis of being a decent person.
Which is why the same people also deny sex, which made them free to be as sexist; and deny race, which made them free to be as racist.
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There's obviously no way to figure this out, but I truly wonder what race relations would be like if all racial agitating was treated like the N-word and limited to American descendants of slaves (and Injuns and railway workers)
It has to be infuriating to hear this shit from not just from ethnic groups that are richer than the white median, but from people like Ilhan Omar who ran cause they fucked up their countries.
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>Some Brahmin whose family had literal slaves in India and was driven to private school in a Rolls lumping himself in with impoverished lead-poisoned African-Americans in Flint, MI as part of the BIPOC struggle.
>"You and I are the same because a teenager once told me my lunch smelled weird."
Or the whole Apu thing when I'd argue the Simpsons targeted the Scottish (Willie) and generic Germanic central European (Uter) more directly with lazy negative stereotyping
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“BIPOC” explicitly excludes dot Indians.
Life can uh, find a way, though.
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We weren't really rich (more middle class) and being black they probably blended in better, but apparently my sister's friends got a bit weird when they realized she was slumming it. Which...I kinda get. We found it frustrating too.
Was it a phase? How did she turn out?
Her run-ins with the law and rampant rebelliousness in high school appear to have been a phase. The damage that phase did to her life prospects (like missing out on college) seems to be more permanent.
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Most people are lying scumbags who'll try to get away with precisely as much bullshit as people are likely to let them get away with.
We have gone farther than this. Being the lying scumbag would not have been a thing in the small town I grew up in.
Now you basically need to be a lying scumbag to just get average opportunity, so everyone does it.
It reminds me a lot of the steroid era and Barry Bonds. He was already a HOF, but everyone else was doing steroids because it had tacit permission from Bud Selig. So he started doing steroids to be on an equal footing.
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