Here's the thing: I bet if you were to take the children who performed well at complex abstract mathematical problems and placed them in a job that required complex applied mathematical skill, the ones who eventually performed best would be the same. And vice-versa for the children who excelled at complex applied mathematics. So mathematical skill might not be immediately transferable, but would be a useful indicator of innate ability.
This relates to the (somewhat controversial) economic theory that much of higher education is not about training or creating transferable skills, and instead is about "signaling". Essentially, we place children in simulated quasi-work settings and use it to determine who has the most of certain economically valuable abilities. By performing well at abstract mathematics, a child signals that they have the requisite mental ability to tackle similarly complex "real world" problems. But also, they signal that they are capable of obeying authority, working with diligence, delaying gratification, etc.
This idea that school is signaling is spread so often, but so much shows it’s not true.
The thing that correlates highest with SAT score is household income.
Household income is also a strong indicator of GPA, school competitiveness, and college attended.
I suspect those with money that like the current system spread this information because it makes them feel their success is their own, and not because it was given to them.
Unless you are a social Darwinist who believe the poor are genetically worse…
Heritability doesn't mean "genetically determined". This is probably the most common message board misconceptions about IQ. The number of fingers on your hands has low heritability; whether or not you wear lipstick has high heritability.
You have to understand what the metric means in order to deploy it in an argument.
It's also significantly genetically determined (twin studies) so in this case I meant genetic heritability.
Also for some reason people seem to think I'm making a comment about high IQ and elitism.
I think low IQ is far more important for poverty discussion. Like sub 80 IQ is a significant percentage of population while being close to or having a mental disability. Sub 100 kind of makes it hard to have a well paying job in today's economy.
And not only is genetic factor big here but also having low IQ/poor parents means your environment will likely have a negative impact on your cognitive development.
No, that's not how this works. You can't just say "twin study" and dispel the environment(s), and, in fact, GWAS is pushing things in the opposite direction (people will argue by how much, but the direction is clear).
Split twins strongly backs genetic component, and heritability is also significant. There's nothing to dispell, environment does seem to be a factor in negative direction, so that's undoubtedly a big factor as well. But low IQ people having low IQ children is going to explain a lot of it - no need to be a "social darwinists" to believe that.
Split twins does the opposite: separated twins IQ-test outcomes track the SES of the families they go to. But "twin studies" is science at the level of "The Bell Curve", which is 30 years old.
At this point, you kind of have to want IQ to be genetically determined (in any meaningful amount) to believe it is.
> Results demonstrate that the proportions of IQ variance attributable to genes and environment vary nonlinearly with SES. The models suggest that in impoverished families, 60% of the variance in IQ is accounted for by the shared environment, and the contribution of genes is close to zero; in affluent families, the result is almost exactly the reverse.
[....]
> In the low-SES group, the intraclass correlation was .63 for DZ twins and .68 for MZ twins, consistent with h2 of .10 and c2 of .58; for the high-SES group, the DZ twin correlation was .51 and the MZ twin correlation was .87, consistent
with h2 of .72 and c2 of .15.
[...]
> In the fractious history of scientific investigations of the heritability of intelligence, the effects of poverty, and the relations between them, there has been only one contention with which everyone could agree: Additive models of linear and independent contributions of genes and environment to variation in intelligence cannot do justice to the complexity of the development of intelligence in children. Only recently have statistical models and computational capacity advanced to the point that less simplistic models can actually be fit. Although there is much that remains to be understood, our study and the ones that have preceded it have begun to converge on the hypothesis that the developmental forces at work in poor environments are qualitatively different from those at work in adequate ones. Clarification of the nature of these differences promises to be a fascinating, and hopefully unifying, subject for future investigation.
Something more specific ? Couldn't find anything directly saying what you're claiming here.
I was interested if my info was out of date so I found this https://www.sciencedirect.com/science/article/pii/S019188692... which basically again confirms genetic IQ factor in adults is major. Separated twins correlate, siblings corelate less and virtual twins (same age unrelated siblings) corelate way less.
Anyway I'm no expert on this, but I've heard this view from multiple credible sources and the more I dig in the more it seems supported.
First one that comes to mind is The Blank Slate from Pinker, but I have read that like 15 years ago. But it did shape my view on the subject and I have seen this topic come up many times since.
I don't remember anyone disputing strong genetic component of IQ, other than people uncomfortable with implications. And I am not even that interested in the social implications, but more practical issues like IQ improvement (basically 0 impact from what I have seen other than being physically fit/exercising), implications on parenting (you cant imprint your desired outcomes on your children and need to play to their innate interests and strengths/weaknesses), etc.
I'm sorry, but this is a persistent myth about this space. Lots of research is done every year on the genetic components of behavior and cognition. The field doesn't see it as a yes/no question; there are questions about how far you can get with classic twin studies vs large-scale GWAS, questions about the malleability of intelligence, questions about the validity of IQ tests (see things like test-test reliability), questions about the meaningfulness of psychometric "g", all actively studied.
Again you can just go read Turkheimer's Vox piece for an refutation of your "basically zero impact" thing.
My big thing here though isn't to convince you that you're wrong about your belief that intelligence is innate and fixed. Rather, it's that you opened this thread with a citation to heritability research, and your usage of "heritability" was broken and misleading. With that cleared up, I don't think we need to drag this out.
> This is guaranteed to be a strong factor, there's plenty of evidence on IQ heritability and correlation with income.
> Just because it doesn't predict the outcome fully doesn't mean it's insignificant.
I'm just trying to figure out what your argument was, since it looks like you're trying to convince GP of what they already believed. Income isn't genetic after all.
The Bell Curve is a great book that is unfortunately maligned by people that dislike what it says but all too often have no actual arguments against what it says.
Separated twin studies show conclusively that IQ is (partially) genetically determined as does, yknow, all common experience and intuition.
Can you name three people who practice in this field (Turkheimer, who I named upthread, is a harsh critic of the book and a giant in this field) who agree with your take here?
> whether or not you wear lipstick has high heritability.
Think about this example. The genetic factor here is massive, much higher than anyone is claiming for IQ. Can you imagine how hopelessly lost you would be if your analysis of lipstick distribution in the population focused on its heritability and link to genetics?
An even bigger issue you immediately run into in these conversations: someone will say "I don't mean heritability that way, I mean genetic heritability". Which, whatever, except: now you've discarded all the science. The papers you'd draw these numbers from are referring to heritability in its technical sense, not in some message board sense.
Yes. I mean, either you're talking about statistical properties of a population or you're talking about something else. These discussions usually devolve into "something else" almost immediately.
You can have intuitions about any of these issues, but you can't use heritability research as evidence for them without understanding the technical meaning of "heritability". That's all that's being said here.
“Regression results suggest no statistically distinguishable relationship between IQ scores and wealth. Financial distress, such as problems paying bills, going bankrupt or reaching credit card limits, is related to IQ scores not linearly but instead in a quadratic relationship. This means higher IQ scores sometimes increase the probability of being in financial difficulty.”
>Previous research, discussed below, has investigated the relationship between intelligence and income and found individuals with higher IQ test scores have higher income.
and my comment says
>there's plenty of evidence on IQ heritability and correlation with income.
>Notice how I cited a source, that’s intelligence. Not just decided I was wealthy and therefore inherently smarter. That’s ignorance.
I will guarantee that you came from a rich family, I have plenty of evidence. Your attitude is strongly correlated with the arrogance of generational wealth.
Please don’t ask me to cite sources, I know exactly what I’m talking about.
If you feel that I’m arguing in bad faith, please consider not responding and instead reflecting on why you feel the way that you do about this topic.
How is anything you just wrote a counterargument to the idea that higher education is about signaling more than about learning?
Signaling that you came from a wealthy family is still signaling. And household income itself may very well be correlated with economically valuable skills developed through childhood that would benefit from signaling in higher ed—that wouldn't make income inequality morally just, but it would make it a positive feedback loop, which is pretty much what we observe.
If anything the idea that higher ed is more about signaling than learning is supported by the idea that the benchmarks we use are correlated more strongly with pre-higher-ed socioeconomic background than with the school you go to. If higher ed were effective at teaching the skills in question we'd see more of a leveling effect than we do.
Perhaps there are different groups of wealthy people with different views on these things?
In my experience wealthy families where the parents are doctors, university professors, and in technology tend to want to transmit a genuine love of learning to their children, as they feel learning and education has done well for them.
On the other hand - my friends are merely millionaires, preparing their children to work for a living. Perhaps there are higher levels of wealth where your kids will never need to work, and the ultimate flex is your kid taking a degree in art history and dropping out half way through because daddy won't mind.
It’s household income of the household they grew up in. Take 10k wealthy kids and swap them with 10k poor kids and you’ll see a massive difference compared to another 20k control
Probably more accurate to say "genetics, shared environment, and non-shared environment". Modern evidence heavily weights (favors? leans towards?) the environments.
Indeed, twin studies form the strongest leg of Bryan Caplan's argument in support of the signalling hypothesis in his book The Case Against Education, which I highly recommend anyone trying to refute it read and try to debunk. If you want an itemized list of citations this is where I'd start.
Hey, it's a free country. :) When I imagine other hypothetical books titled things like The Case Against Capitalism, The Case Against Monogamy, or The Case Against Atheism, though, I note that I don't get an "insider propaganda" vibe from any of them, even though I would probably strongly dislike what they have to say.
Dr. Caplan in fact does cover this point in TCAE, of course. He comes to conclude that only about 70-80% of the effect of education is attributable to signalling. A solid 20-30% still looks like good old fashioned human capital improvement, and it is largely concentrated around the basic primary education skills of reading and arithmetic. (He even has the spreadsheets where he calculated all this out online, and he has talked before about how sad he is nobody has ever tried to fiddle with the actual numbers.)
Probably not actual "2a+4=12, how much is a?" style basic algebra, though. In the United States, which is about lower-middle of the pack on PISA, about 1 in 3 adults would struggle with that level of algebra according to the PIIAC, to say nothing of e.g. the actual compound interest equation, even if the rough idea makes sense.
That's not "most people", but it's definitely "a plurality" of people. And yet life is pretty great!
My last point is that life in America is pretty great. You don't deny this, to your credit. But I don't see why that would link to "American democracy is threatened". If anything I would expect the opposite to be true.
"There's a threat to American democracy" seems like a strong claim to me by itself, let alone "There's a threat to American democracy partially because of its education quality." But, I'm an American myself, and I don't want to play inside baseball with how likely that actually seems to me.
Let's instead take Germany, where you yourself seem to be located. Germany has PISA scores quite close to the US's own, maybe slightly above or below depending on which recent year you look at.
If poor education leads to collapse, and if the two countries are about equal in their poor education, you should then be willing to accept, say, a 1 to 20 bet that German democracy will itself self-immolate in, say, 15 years. But, if poor education doesn't justify even a 5% risk of this happening in Germany, then I don't see why I would think it's a relevant factor in predicting the collapse of democracy in another country with a much longer uninterrupted democratic tradition.
(You could of course argue "No, comparisons based on PISA scores are misleading, actually there's robust pro-totalitarian brainwashing happening in US high schools that doesn't happen in German gymnasiums", or something, but (a) that's a much more precise claim than merely "US education is bad", (b) that seems really unlikely to me given I've never actually met or had an openly pro-fascist teacher at any level, and (c) even if it was true, the signalling hypothesis would still suggest any attempts at this just wouldn't matter very much by the time these kids are 25 or so.)
There's nothing mean-spirited about asking people to put rough numbers to their beliefs, even approximately. But I do think most people would be genuinely surprised to hear you think there's over a 5% chance Germany the Western democracy won't exist in a generation. If it does happen I need to remember to revisit a lot of things about how I myself understand the world.
I for one like books which are willing to explicitly make a controversial point; it makes for much more interesting reading, even if I remain in disagreement throughout.
With all else being equal, I'm willing to believe that more intelligence would lead to greater income. But if you're claiming that IQ is the sole/main predictor for income that'd be a hard [citation needed] in my opinion.
In fact I recall a study (I saw on HN I believe) that showed that IQ is only correlated up to a certain value (not very high, I.e. Lower upper middle class IIRC) but then becomes quite uncorrelated. This certainly matches my anecdotal evidence that most of the rich kids during school were not very smart. The smart ones were typically the kids from academic middle class households.
> I don't find it too surprising that people who are smarter make more money.
Yeah, being smart definitely helps with making money, but honestly, that bar’s not as high as people think. There’s a bunch of other stuff that matters too, like being likable and humble. But if we're being real, you’ll probably get richer with cunning and greed. And history is pretty clear, the more opportunistic ones tend to stack higher piles. Money is cruel, man.
those with success like the idea of meritocracy because it means they deserve it, makes it easier to deal with the injustices of the world. very very rarely will someone at the top of their field say "i was lucky"
ish, yeah, but when you go work at a job it seems like half the people there are because they're smart and half are they're because they're upper-class/wealthy background, and some are there because they worked really hard, and a few are there by accident. at a daily experience level it's very clearly more complicated than smart=money; it's just one weight out of many.
Poor people aren't necessarily genetically less academically able. They generally have a weaker academic nurturing at home and often in school, which leads to lower developed ability. This is not controversial. What's controversial is how much and what kind of support they deserve from society to overcome the disadvantage in environment.
> What's controversial is how much and what kind of support they deserve from society to overcome the disadvantage in environment.
I think calling that controversial is taking a side. We're talking about educating children, who are innocent, and greatly increasing the welfare and productivity of society.
If people argue over it, then it's controvercial. It doesn't matter if you think it should be or not. It doesn't matter how valid you think any of the arguments are.
That's one side's argument. You can argue with anything, and then it's 'controversial'. Round Earth vs flat? Controversial! Climate change? Controversial! Basic economics? Human rights? Freedom?
If there are two big sides arguing then that is the definition of controversial. Uncontroversial means there is no big opposition to it.
> You can argue with anything, and then it's 'controversial'.
Yes, that is the definition of the word. I am not sure why you try to define it as something else, something being controversial just means people don't agree about it.
Taking a side? I live in Portland. We passed a tax to make preschool free for all children. Rich people fought it tooth and fucking nail. This one thing that can have such a huge positive outcome was wildly controversial. It passed but man they are still fighting to repeal it. So calling it taking is side is wildly ignorant of the facts. And this is in bright blue Portland, not some red state educational desert.
Out of curiosity, why do they fight ? Is it because they don’t want to pay for it, and will put their kids in non free pre schools anyway ? ( I’m from Europe so the idea of fighting against free pre school for everyone sounds a bit odd )
We'd have to find surveys to know that. One possibility is that it's reactionary politics: they are against liberals and attack them in every way possible. For example, being against car electrificiation, climate change as a fact, Covid vaccines, wearing masks, DEI, etc. It is the fundamental ideology of the right wing, it seems.
They don’t want to pay another 1.5% tax. I guess if you’re kinda rich but not super rich you can’t avoid taxes the way super rich do. So this just pisses them off because they are just upset with taxes. I dunno if I made that much money seems like it wouldn’t be that big of a deal but I’ll never know. lol. It is just one city doing it and for just that city. I can see their point and maybe if the tax was smaller and applied to everyone we could all contribute but with the cost of living so high and how helpful preschool is for working families … people just don’t seem to like to help strangers that are worse off in America. It’s Everyman for themselves and their immediate family.
There's some evidence that preschool improves children's educational outcomes in the short to medium term, but actually is negligible or even harmful in the long term.
A study of Tennessee's pre-k program showed that students who went to pre-k outperformed students who didn't in first grade, but by sixth grade, they performed worse.
This article says more research is necessary and I cannot read their methods so it is a bit disinegous to cite this as some kind of proof. Especially when the article says early education programs vary in content. Tennessee has one of the worst school systems in the US, #41:
Tennessee was the most stark example, but not the only one. And the fact that education is often sub-standard isn't a good case for more of it.
The point: it isn't blindly obvious that devoting even more of our children's lives to an education institution is necessarily best for them, in all cases.
> The thing that correlates highest with SAT score is household income.
Something like 25% of students at Harvard had a learning disability, one of the highest rates in any school in the nation. Do you think that's real disabilities, or their parents know doctors to give the diagnosis, admissions coaches to tell them to get one, etc. With the learning disability the student gets unlimited time on the SAT and can plug and chug all the math answers instead of having to use heuristics of eliminating etc. On the reading comprehension they can read the passage five times over. On writing they can turn in their 8th draft.
I grew up in one of those small hometowns that sends a lot of people to the close by Ivy League school.
It was an open secret that getting a learning disabilities diagnosis is a great SAT booster. Also makes you eligible for more time on regular school exams and you can use it to later get more on the LSATs (not sure about other grad level exams, but I am sure it’s similar)
You missed the 3rd option which is that (most) of the disabilities are real but the accommodations we have massively overcorrect and opens up new strategies that weren't possible when you're given intentionally less time
than you need. So you get students who come from an environment outside the "normal" gifted path of private college prep schools who come out of nowhere and score amazingly well. That combo sounds like catnip to an admissions office.
I scored really well on my tests, a fact I credit mostly to my adhd (I did not have accommodations, brain goes burr under time crunch), so I have every incentive defend the current system but looking back I genuinely don't understand what we're actually testing for. Perform! Under pressure! In a completely different environment than you're used to, while proctors watch you like a hawk. Your whole future rides on this! is not how I would describe a wholistic assessment.
> In a completely different environment than you're used to, while proctors watch you like a hawk. Your whole future rides on this! is not how I would describe a wholistic assessment.
I would wager that a large portion of Harvard admitted students did not in fact have this as an unfamiliar environment for the first time, but had like a whole year of test prep and simulated tests and took the tests multiple times if they weren't going to get in with their initial result, which could be a money barrier for others.
The argument that schooling is signalling wasn't as true in the past as it is now. In the past (think mid-20th century USA), everyone did not have ubiquitous, instantaneous, essentially-free, 24/7 access to more or less all human knowledge ever recorded, from more or less anywhere on the planet, using a rectangle that fits inside your pocket. Back then, information was comparatively scarce, so stockpiles of information had (again, compared to today) much more value to someone who was strictly interested in the information.
These days, people who want an education strictly for the information don't have to put up with any of the crap higher educational institutions drag their victims through in order to procure that information.
Anecdata: me. Last year I had a gross individual income that placed me within the top 1% of US income earners. After a little under a decade at a MAG7 company, my side project, starting a quantitative hedge fund, has progressed to the point of providing me with more income than my MAG7 employer was. All of my tech and finance skills were obtained for free online.
This is the same me who did not even take the SAT, who dropped out of a public community college, who's highschool GPA was likely in the low 2's, but even that was inflated.
I'm also debt-free and financially independent before 30. I went to school with some kids who made it into ivy leagues. Most of them are earning high five or low six figures, many still have student loan debt.
Also, make no mistake - the loudest proponents of higher education feel the same way about the uneducated as the social darwinists do about the poor, as well as talking to and treating the uneducated about as well as social darwinists do the poor. There's no moral superiority here, they're both priests of delusional, artificially constructed social hierarchies designed for the belittling, ostracization, and exclusion of others.
In a society that leans more towards merit, where the highest compensated jobs correlate with higher IQ and analytical reasoning ability, then incomes and SAT scores will correlate.
That said there are some total dumb fuck rich kids, and absolute genius poor kids. Just because there is correlation shouldn’t imply anyone is destined anywhere.
But we're distinctly not in a society that leans more towards merit.
From what I can see, the highest compensated jobs go to morally and ethically bankrupt raging narcisists, successfully masking psychopaths, and spineless puppets for the previous two categories.
There's a lot of it about.
(And IQ is not a particularly useful metric of overally intelligence. It's a useful metric of measuring how people perform on a common set of problems).
and also points out that wealthier people also do better on everything else. Standardized tests get talked about because they're visible, but there are all sorts of things that aren't visible such as many forms of opportunity hoarding (wealthy parents will push harder to get their kids into gifted programs, can afford to get kids out of toxic environments where they get bullied, can put use the carrot and stick on teachers to get boost their kids GPAs) etc. For example there is a low level of awareness that certain sports like Polo or Lacrosse at the Ivy League privilege rich kids because they are more likely to play that sport in school.
Thing is, everybody knows the SATs are important, but there are 100x extracurricular things you can do that wealthy people know about and poor people don't that get only a small amount of criticism because they're invisible.
Come on. There are no sources that would report this. It makes as much sense as claiming that "as everyone knows, the greatest contributor to tidal pull is the odor of the moon as its cheese rots, which attracts plankton on the ocean's surface". And it has as much scientific backing. Where in the heck are you getting your "facts" from?
Here, from a book chapter subtitled "The case for eliminating the SAT and ACT at the University of California":
Table 1.1. Correlation of socioeconomic factors with SAT/ACT scores and High School GPA
+--------------------+---------------+------------------------------------------------+
| | Family Income | Parents' Education | Underrepresented Minority |
+--------------------+---------------+------------------------------------------------+
| SAT/ACT scores | 0.36 | 0.45 | -0.38 |
+--------------------+---------------+------------------------------------------------+
Of the three factors they bothered to look at, family income had the lowest correlation, just below a poor approximation of race and very far below parental education.
I looked at that table and immediately thought, "Those are pretty solid correlation values", so I was confused by the point you were making.
I also read the page in the PDF you linked that was the source of the table and became more confused, because it very clearly writes out that:
>What this means is that 40% of the variation in students’ SAT/ACT scores is attributable to differences in socioeconomic circumstance.
I think you are either:
1. Misunderstanding correlation coefficients. Correlation coefficients range from -1 to +1, implying a 100% negative or positive correlation. 0.4 and -0.4 are moderately positive and negative correlation values. That table is saying that if you are an underrepresented minority, you are expected to do 40% worse on the SAT/ACT.
2. Quibbling over the "thing that correlates highest with SAT score" claim, because you feel like 0.4 is not high enough of a number. The only factors with higher correlation values than 0.4 in SAT/ACT scores would be stuff like "Was the student currently injured?" or "Did the student make it to the testing center?"
> That table is saying that if you are an underrepresented minority, you are expected to do 40% worse on the SAT/ACT.
This is separate from the main point, but "underrepresented" minorities are defined by their lower performance on the SAT, so 0.4 isn't impressive at all for that cell.
Also, you've made a gross mathematical error; "you are expected to do 40% worse on the test" is a statement about the effect size, not the correlation.
Of course if having a brain would be on the table there would be no contest.
Being the highest correlation in a rather arbitrary list is not a valid qualifier for being a good correlation.
> Of course if having a brain would be on the table there would be no contest.
Is that really what you think? In reality, the possession of a brain can't have any correlation with SAT scores at all, because there is zero variance in the trait.
And parental education, race, and household income are also all correlated with one another... And the paper you cited makes the point that the correlation between socioeconomic background and test scores has increased substantially since 1995. This is not the gotcha that you think it is.
> And parental education, race, and household income are also all correlated with one another...
That's true.
> This is not the gotcha that you think it is.
But your point makes the guy I'm responding to look much worse. After controlling for parental education, the effect of parental income is radically diminished. Most of that 0.36 correlation is just confounding. Parental education isn't even a great indicator itself; it's a proxy for intelligence. It's just a much better proxy than parental income is.
Signalling, and simple competition. School is also about assigning numbers and ranks to students, labels that control access to higher levels of education. Collecting the correct numbers and badges opens doors irrespective of whether they are a marker of ability.
Just going from memory, but something in support of that is that when a country puts out more people with advanced degrees, this does very little to increase GDP. It does much less than how much an individual gains in income with an advanced degree.
This hints it’s not actually making people more productive. It just lets them get a bigger piece of the pie.
First, in many case, beyond just math, academic instruction does not directly apply to practical application. But, the reverse is absolutely true. Experience with practical application very closely correlates to better academic performance. The key that you missed is practice. Instruction is helpful, but it isn't practice. Practice is also the key difference between graduate education and high school or undergraduate education. Yes, in graduate school there is still instruction but you are expected to use that instruction in more abstract ways to achieve some condition.
Secondly, higher education is mostly about signaling. It is about other things too, but signaling is first. Examples include the social consequences of where you go to school, the major you choose, grade point average, who you meet, and so on. Compare those factors against professional education, like continuing education credits for law or medical licensing. Nobody cares about the signaling around that professional continuing education, because you are already employed and the goals of that signaling are already achieved.
I don’t know much about any of this but I know that the best engineers I’ve worked with all went to community college or nothing, and the worst engineers had PhDs.
But if the job was to invent some brand new algorithm and not to build a product and get it to market expeditiously, maybe that would be backwards. I dunno.
In the United States, the term "Professional Engineer" is a legal term describing persons licensed by their state. [1]
The generic title "engineer" is not legally protected and may be applied to software developers or many other professions where scientific understanding or invention is applied. For example, software developers at Amazon and Microsoft are called "Software Development Engineers".
Most electrical and computer engineering graduates do not bother with getting PE licensure; instead this is a requirement for civil, mechanical, aeronautical, and related engineering graduates.
But honestly, the whole point of the original study here was testing something that sounded correct, and it proved to be incorrect.
So I have the same hypothesis as you, but the original study here shows the importance in testing things regardless of what sounds correct. It's way too easy to spread overconfidence using a theory which matches passes people's people's smell test.
I think this impacts mathematics education particularly acutely. Math is an area where theory rules above all else and many people working in the area apply theory to problem solving. But mathematics education is not mathematics. It is not theoretical. It is social science. Experimental evidence and testing is the way to make progress - not sitting back and thinking.
I'm afraid that this is because academic math is often taught and tested in a way that rewards memorization rather than understanding. Here's Richard Feynman's take on the problem:
I had a similar reaction. I had a lot of reactions, and found the paper interesting.
First, it reminded me of something a stats professor said in grad school: "there are two kinds of mathematicians, those who are good at arithmetic, and those who are not." He was speaking as someone who identified with the latter.
I can't tell if this is something related to this domain of math in particular or something broader. My guess is it's something broader.
I have colleagues (speaking as a professor) who have complained about admitted students who come in with very high grades and test scores, but who can't actually reason independently very well and despair when they are not "told exactly how to respond" on tests and whatnot. You have to be careful because sometimes these complaints hide bad teaching, but I think this is a common sentiment, and I've seen articles written about similar sentiments at other places.
The paper touches on a lot of issues, like applied versus abstract concepts, generalizability of learning, "being a good student" versus actual cognitive ability, learning how to take tests versus learning concepts, the difficulty of measuring cognitive and academic ability, and the fallibility of measuring complex human attributes in general.
Even in lower education, as a student I hated word problems. Partly, I just wanted to be told what equation to solve. In retrospect, though, I think a lot of it was the framing.
It was always presented as some variation of short exposition followed by a question. The question was usually framed as an outside observer asking for some fact about the story.
Think of the classic "A train leaves station A headed west at 6:30 traveling at 30 miles an hour. A second train leaves another station at 7 traveling 50 miles an hour. When do they pass each other?". There's no problem here to solve. Who cares when they pass each other? Why do we care?
Sure, a little exposition helps build up analysis and application skills, but it doesn't actually offer much in the way of engagement.
I was a college math and physics major, and much later taught a college freshman math course that was a level below calculus.
The point of word problems was to recognize a pattern matching one of the topics from the latest chapter, fill in the parameters, and grind through the memorized algorithm. As a student, I liked word problems, but I knew the secret. It was all a game.
What made math come alive for me was proofs. As for applied skills, I developed those in the lab, and making things.
True, looking at real proofs is what changed the game for me.
Before I actually went through 3-4 books on basics of proofs, math felt... almost meaningless , a game of remembering the right thing at the right time.
Saying that as somebody who oscillated between being "good in math" and "top in class" for all 18 years of studying.
To me proofs never where the interesting part of maths - the ideas and intuition which made the proof possible were.
Proofs were a way of formalizing something and, well, making sure the intuition was actually correct, but they were just a tool and not the game itself.
The best math teachers/professors I had were the ones who focused on the ideas .
Yup, and once again, it depends on how we learn. I'm a strongly "learn by doing" kind of person. For instance, I'd get almost nothing out of reading a math book that was full of ideas but no problems or proofs. Doing problems and proofs is how I wrestle with the structure of the subject matter, and internalize the ideas.
In elementary school, I hated word problems because I kept thinking of things that weren't specified which prevented there from being just one right answer. Sure, the car left City A at 60 miles per hour, but what if there's a stop-light? I know there are lots of stoplights, so it must go slower, and you didn't tell me how much slower it would go...
I like to think that I've turned it into an asset when it comes to software. ("We don't know that the first parameter won't be null...")
"Why should the reader care?" I agree, they aren't framed in a way that engages the reader.
How about for a division problem, start with a bag of candy, or if it HAS to be healthy, a bag of cherries.
Or maybe apply it to cooking. Lets use Metric anyway, even after ( https://en.wikipedia.org/wiki/Metrication_in_the_United_Stat... ) and ask questions about a recipe for some food dishes (use real ones! IDK maybe bread, pasta, some pastry stuff...) and ask things like the total expected volume based on the ingredients. How much X there should be if naively adjusted by exactly a factor of 1/2 or 3x etc. Things people might do if a thing was intended for a family of 4 rather than 2, or a group of guests at a holiday.
> There's no problem here to solve. Who cares when they pass each other? Why do we care?
Not trying to “but acktually” you, however, this is more or less how I calculate the optimal time to take a pit stop in a lap-based auto race. I have a little spreadsheet widget that I made to be able to plug the numbers in, but the problem is simply stated:
If old tires decrease my speed, and making a pit stop takes time, when should I stop.
Agreed that elementary school word problems are dumb, though.
The original problem, and the racing one, are both logistics problems, and everything in the world runs on logistics, and people have to be good at it.
If you don't like it or aren't good at doing it even while not liking it, the problem is not the problem.
I don't think you are arguing against the parents position, but for it, while your answers' odd contrarian positioning also exhibits how critical context and caring are to answering questions. Good job, you.
I think you've missed my point. There's no interesting logistics problem in "when do the trains meet?" Yes, the underlying math is useful for all sorts of things, but the word problem doesn't offer any motivation for knowing the answer. It's purely asking as an impartial observer.
Back when I took calculus in high school, my teacher explained how traffic speed cameras used mean value theorem to prove a car exceed the speed limit.
Here, there's an actual problem, actual actors and observers, and a motivation.
It answers the question why is this useful to know, or to be able to answer.
> There's no interesting logistics problem in "when do the trains meet?"
This is trivially resolved with "and the first train is carrying an urgent package for a passenger on the second train. When will the trains meet to deliver the package?".
But on the exam form, that's just extra irrelevant noise.
Applied math in daily life, such as making change, also involves a lot of memorization, it's just that the person doesn't realize they're committing various formulas and equalities to memory.
The problem with proficiency in, e.g., making change, is that it doesn't carry over to higher-order math, logic, and reasoning. Arithmetic as taught in elementary school is attempting to achieve two things at once: proficiency in applied arithmetic, and foundational number theory (e.g. commutativity).
My mother was a waitress and emphasized skills like making change. While she never articulated the rules, I ended up developing many of the mental arithmetic techniques that (I later discovered) Isaac Asimov discussed in Quick and Easy Math: https://archive.org/details/QuickAndEasyMath-English-IsaacAs... But I never developed an appreciation for number theory until it was too late--i.e. after high school. I did get into philosophy and logic during high school, but the connection (theoretical and applied) between the two didn't click until later.
Sadly (or not?), proficiency in mental arithmetic has become much less common even among waiters, clerks, etc, at least where they don't deal in cash directly and without the aid of a register. And professional mathematicians have always humble-bragged about their impoverished mental arithmetic skills. So maybe we should drop the pretense that we're attempting to teach applied mathematics in the early years and admit the purpose is to lay theoretical foundation for higher-order math, applied and theoretical.
I only learned this in adulthood, too. I had to discover and memorize those shortcuts. Now I know that some of the "brilliant" math students I went to high school with had simply learned these skills the rest of us didn't.
That take has been accurate in specific times and places, but it is also blindly repeated in contexts where it's simply not true. All the teachers I encountered in my time in the US public school system were trained in the progressive spirit expressed by Feynman, with a strong bias against rote learning and in favor of a conceptual, understand-based approach. My math education in school both encouraged and rewarded figuring things out, which was good for me because I hated memorization and was always bad at it.
Despite that, the criticism that school rewards memorization and doesn't teach critical thinking is still the only criticism I ever heard about the education I received. It's the standard thing that well-meaning people say.
Which I think is a shame. When virtually every teacher in the system is trained in the progressive approach to education, and most of them sincerely believe in it and do their best to practice it, only to have the entire society turn around and claim that they are actually implementing ideas that nobody has believed in in a century, must be incredibly discouraging.
Yeah, it seems almost as if a lot of people look back at their experiences of U.S. schools in the 90s, and assume that the schools in 2025 must not have changed one iota. While obviously I can't speak for every school district (nor can anyone), many of the criticisms I've heard seem disconnected from the schools I'm familiar with. (Not that they aren't subject to newer criticisms!) Is my personal experience a big outlier, or are people just extrapolating from the past? It makes me worry that school districts will greatly overcorrect in their efforts to ward off the old criticisms.
I went to school in the early 1990s! The progressive approach to education was already orthodoxy when my teachers were trained. It has been around a long, long time and has been the prevailing belief in education for half a century or more.
The situation is almost paradoxical: you have generation after generation of people saying that education needs to be reformed to eliminate rote learning and focus on understanding concepts, and where did they learn this orthodoxy? In school, from their teachers.
I suspect it has something to do with how teenagers experience school. No matter the pedagogical approach, if kids are distracted with their social lives and normal adolescent stuff, they experience any attempt to teach them as dry and rote.
Feynman was correct for science in school, however arithmetic is fundamental and maybe one level above the root of all mathematics. Children should be able to do most of it via mental lookup tables and apply that knowledge on paper. For some reason, they can't.
No, it goes beyond that. There's "arithmetic," the applied usage of addition, multiplication, subtraction, and division to permute numbers, and then there's Arithmetic, the set of theorems and axioms that give rise to that system of applied arithmetic. Memorization only works for the applied part, and children aren't usually taught that there is a system of reasoning behind those rules. Without that, no amount of mathematical dexterity in pushing symbols across a page will help them understand anything past the 100 level, and sometimes not even that.
I also think there's a huge undercurrent of resistance from adults to having children learn that system of reasoning because adults don't understand why it's useful, and in my experience when people don't understand something they dismiss it.
Edit: A nice example of another axiomatic system that might be more approachable is Euclid's Elements, in which five postulates are used to develop a system of geometry using an unmarked straightedge and a collapsible compass that you could, if you were careful, use to build bridges and other large buildings.
Once I got to calc2 and 3. I was so mad. I realized I had spent nearly a decade memorizing things. When I could use calculus to have a factory that made formulas and the rules were on a whole simpler to remember and apply.
I had a lucky experience taking HS calculus the semester before as physics. I saw other students torturing themselves memorizing the formulae from the physics text and even then struggling to apply it to novel problems.
For the most part, knowing basic calc, it was possible to just draw a free body diagram and either integrate or take a derivative to get the answer. Didn't memorize much beyond f=ma and v=IR, for better or worse.
I still firmly believe that physics and calculus should be introduced together to provide a tangible and practical base to understand the mathematical theory.
It's common in European universities to not have "service courses" that the math departments provide to other departments. The other departments teach the math that is part of their field!
Similar here: There were all sorts of volume and area equations I could never remember, then one slow day at work I decided to try and derive the volume of a spehere using what I'd just learned in calculus. After doing so each part of the equation made sense instead of appearing random, and two decades later I still remember it without having to derive it again.
I mean I remember seeing this first-hand student teaching Mathematics 13 years ago in the US. They got to me having never seen any of that stuff, and the curriculum attempted to provide a good education in mathematics. But the staff and the way the whole system is structured is to skip all of that and memorize the single rule you need to know to get through the test. So it's all done by rote and the only time you find out how you've been cheated is when you try to go through Calculus.
And I remember that was how we learned everything when I was a kid, and the teachers chose not to do anything else. I also remember from my math ed curriculum one of the professors joking about the elementary education students complaining about having to learn middle school math from the college perspective. So I think portions of this apply here.
I've also seen carpenters apply trigonometry very effectively to do things like cuts for roofs and stair jacks, so there's certainly a lot of truth to people learning maths by occupation and not in a formal setting, and I think part of it is the formal setting.
Why should they? We have the tables in our pockets at literally all times; doing arithmetic without it might be useful, or a bit faster sometimes, but is hardly an essential skill.
It builds a numeric intuition. When you repeat something enough, it begins to do itself - you gain a subconscious mastery. Think about yourself as you read these words. Imagine if you were looking at the letters and actually trying to sound out each word, consciously thinking about each words meaning, and then finally trying to piece together the meaning. You'd spend 5 minutes reading a sentence or two, and oh God help yo if tere ws a tpyo. Instead it all just flows without you even thinking about it, even when completely butchered.
And that sort of flow is, I think, obtainable for most of anything. But 100% for certain for numbers. Somebody who doesn't gain an intuitive understanding of basic arithmetic will have an extremely uncomfortable relationship with any sort of math, which mostly just means they'll avoid it at all costs, but you can't really. I don't even mean STEM careers, but everything from cooking (especially baking) to construction and generally an overwhelming majority of careers make heavy use of mathematical intuition in ways you might not consider, especially if you're already on good terms with numbers.
that's why montessori math is so impressive. it starts with counting out beads one at a time by the hundreds until they have internalized that. then they get beads on a stiff wire 10 at a time, and repeat the process counting them out up to a 1000. and so forth until eventually they hold in their hands blocks of 1000 beads glued together in a cube, and only after they have internalized that the beads get replaced with more abstract woodblocks and sticks. and all that happens in the first year or so at the age of 3.
It's a bit ironic to be saying this in the context of HN, the tech stack is built up on layers of abstraction that little of us have mastery of.
If we made it so that only people who mastered assembly could be considered "real programmers" we'd get nowhere, certainly not to build modern web applications or video games.
People that have a poor understanding of the stack under them almost invariably produce very poor software.
Video games are an area where in fact good software is still produced, mostly because the people working on the cores of games DO know (at least how to read) assembly language.
Modern web applications on the other hand so basically the same things we were doing with computers 20 years ago but consume 100x the resources to do so.
No "modern web application" comes anywhere near the quality of Word or Excel 2003.
so every time i go shopping i have to type all the prices of what i buy into my phone and also have the calculator connect to my bank account and not only make sure i have enough savings, but also tell me that i am not spending more than my average for weekly groceries? and when doing that i need to make sure to not make any typos because my lack of numeric intuition won't allow me to recognize where i made a mistake. and i also won't be able to tell if an item is overpriced. nor will i recognize a bargain unless it is marked with a big colorful sticker.
I struggle when trying to solve math problems without context. I learned enough trigonometry to pass the final exams in high school, but I didn't REALLY understand it until I took a graduate-level graphics programming class.
Some people enjoy the process of solving equations and math problems. For me, it's a tough process. Unless I have a tangible goal, I struggle to visualize the problem.
Starting with basic algebra, it would be more effective if mathematics were paired with some practical problems. Computer graphics, engineering, construction, finance and the analysis of data would be good areas to do this in because it's exactly where you'd need said math!
I wonder if this is true for all cohorts. There are a lot of children who are just fundamentally not intelligent, and deal with math classes by basically memorizing things and repeating them without real understanding. But for children who are understanding what they’re learning, I would expect academic learning to translate to other things.
I agree. Academic math is taught as a set of rote rules or steps. The focus is not on intuitive understanding. I was taught the usual method of long division and carrying all by rote. Only later on in my academic life did I work out on my own why it works as it does.
This hasn't been true in most of the USA for decades.
A common failure more is for students to forget something and then claim they were never taught it. Arguably the should have been taught it more thoroughly.
i hate when people quote random celebrities as authoritative on any topic, let alone as a counterpoint to actual authorities (google the authors of this study).
Edit: hn is just as anti-intellectual as any other place these days but y'all style yourselves as intelligentsia because your celebrities are special.
I'll repeat: check out the qualifications of the authors of this study and compare them to Feynman's on this subject. Any reasonable person would conclude that comparing them is exactly like comparing Kim Kardashian and Feynman's on QED.
He is celebrity. He is not authoritative about subject of teaching math to kids. He is authoritative about his area of physics. He also is authoritative about writing popular books for physics that demystify physics to adults. But again, not about kids and math.
"you know him because he's known but because you're actually familiar with his work"
I cannot parse your statement, either there are some missing words, or some problem with your English.
Anyway, from my previous comment, you couldn't have any idea about how I got to know Feynman and his work. I haven't mentioned it at all.
FYI, I got acquainted with his work in 1996 when I enrolled into the university. I was studying maths, my dormitory roommate was studying theoretical physics, and he had several Feynman's books that were very interesting to me, though I must admit that sometimes the underlying apparatus was really complicated for an 18 y.o. greenhorn. But the principles were clear enough.
Linus Pauling's authority in chemistry doesn't make his cuckoo theories about Vitamin C any less cuckoo. Feynman may have been an important physicist, but that doesn't make him knowledgeable about education!
And, to be honest, there's a reason why there are memes about physicists' competence in other fields, like https://xkcd.com/793/.
As well as the texts based off his lectures. [1] His ability to teach was completely unreal. Those 'books' dramatically deepened my understanding of physics.
This is not quite accurate. I mean it is, but only in the same sense that a top 5 chessplayer in the world might regularly bemoan, with no irony intended, his inability to play chess well. There's a lot more context to his comments here. [1]
Seriously, if you are at all interested in physics - read the lectures and they will, with 100% certainty, deepen your understanding. Even on the most fundamental topics. For instance my entire worldview around the conversation laws changed thanks to those lectures, which in turn ties directly into the nature of energy.
Probably because a lot of us have read or watched his work and know first hand that it was extremely high quality. It's not like you have to take people's word for it.
The Feynman Lectures on Physics was used as the textbook for Caltech's introductory physics course for nearly two decades, and it is still used in some universities. I learned physics from it and have met many Caltech alumni who used it as their textbook, all of whom felt they learned a great deal more than "intuition" from it. So I am guessing you've never actually tried to learn something from it if you feel that way.
He taught a two-year introductory physics course at Caltech from 1961 to 1964, which gives him some experience with the matter though. He was known as "The Great Explainer", due to his ability to help people understand and more importantly, be inspired by science and the world around them*. His materials from those lectures were converted into "The Feynman Lectures on Physics",
a highly regarded physics textbook. so I wouldn't have chosen education as my example.
In support of 793 however, he didn't do well with bureaucracy so I'd not listen to his advice on how to run something that favored rigorous rule following even when the rules don't make sense**.
My parent comment is especially jarring because Feynman's findings agree with, and propose a mechanism for, the findings of the study. The comment seems to imply that there's some great tension between "arithmetic skills do not transfer between applied and academic mathematics" and "the students had memorized everything, but they didn’t know what anything meant". Or between "These findings highlight the importance of educational curricula that bridge the gap between intuitive and formal maths" and the less academically worded "There, have you got science? No! You have only told what a word means in terms of other words. You haven’t told anything about nature."
Nobody's quoting Feynman "as a counterpoint to actual authorities". Feynman's excerpt provides first-hand testimony from a teacher on the front lines that fully validates what the study found.
Feynman is no random celebrity. In addition to be a renowned physicist, his famous "Feynman Lectures" and his thoughts on pedagogy are similarly legendary.
The Feynman Lectures are great at giving you an intuitive understanding, but is no substitute for the regular curriculum. You don't find many people who read only the Feynman Lectures who can then go on to solve physics problems well. You do find many who read the regular textbooks and who can.
You have to bear in mind that the lectures in The Feynman Lectures on Physics were only one third of an introductory physics course, the other parts being recitation sections (in which homework problems, quizzes and tests were given and discussed), and labs. Lecture attendance was optional - many people prefer reading to listening - but the recitation sections and labs were mandatory, because they were considered much more important. Nobody learns physics from just reading lectures.
In this case, Richard Feynman is just writing about his personal experiences of a well-known phenomenon. https://profkeithdevlin.org/wp-content/uploads/2023/09/lockh... ("Lockhart's Lament") would perhaps be a better reference, but nearly anyone who's been through the education system would be able to tell you this.
And why should I simply assume that "Education Economists"* really know the subject they purport to talk about? Because they are credentialed members of university departments with some label? Because a few of them won some Bank of Sweden award?
Just because a particular department or field of study exists in academia does not magically give them the imprimatur you think it does.
* Btw, I know for a fact that a few of them are not "education economists"
Richard Feynman is famous for being an educator, and he's clearly quite good at it. Who cares if he has no formal training? I reckon he deserves at least a 1.2 on this scale.
It's amazing how deep the celebrity worship goes. No he's famous for being a mathematical physicist (his Nobel is in physics not education). He was actually a very mediocre educator - you can read his own assessments of his success/failure in teaching the "famous" intro courses.
Or you can ask literally any physics major that's actually had to use those books (they are horrible for actually learning from).
I wanted to upvote your other comment because it caught a detail of "how much" that may have slipped past the other commenter's or other reader's minds but...
0. The Kardashians
The distance between 0 and 1 is vast compared to the distance between 1 and 2. Feynman was a professor and also beloved for his ability to bridge across the academic to pragmatic divide that is the subject of this paper.
What is the relevance of this point? No one has linked a Kardashian's take on anything? So who cares if the distance between 0 and 1 is larger than the distance between 1 and 2 - we are only discussing the distance between 1 and 2.
The original comment you responded to made no comparative claims. It simply offered another person's attempt to describe. Feynman is fairly famous but nonetheless an authoritative source relative to most of the population (probably more so than both of us, though I don't know you do have little basis beyond priors [sorry if you have greater credibility than Feynman, I didn't know]). Feynman is less authoritative on the subject than the authors of the article but still... Being well known doesn't remove the authority level that Feynman does have on the topic.
It's not a counterpoint. The Feynman excerpt and the paper support each other.
The paper's abstract ends, "These findings highlight the importance of educational curricula that bridge the gap between intuitive and formal maths."
The Feynman excerpt is about the issues caused by a lack of practica in education and how they should be resolved.
The paper's authors wrote, "These findings call for a maths pedagogy that explicitly addresses these translational challenges through curricula that connect abstract maths symbols and concepts to intuitively meaningful contexts and problems." And provide 2 examples of Randomized Control Trials of math courses in Brazil and India respectively that address the challenges successfully.
Even if you remove Feynman's name, it's still interesting that a Theoretical Physics professor and educator wrote clearly about a very similar issue they encountered over 60 years before the paper in question was published.
Regardless of the people involved, being asked to consider someone’s opinion on a matter is a world apart from claiming they are an authority on the topic.
I run a microschool where I teach math (and I am a neuroscientist) and this is pretty obvious, and also something we see every day. And it's because we still suck at understanding learning.
Learning is not the acquisition of knowledge. Learning is all the things by which we learn to model the world. And we do that, our brains do that, because it matters.
Math in classrooms is pretty much designed to leach out all context. Meaningless symbols that need to be manipulated to arrive at some mysterious answers. Why bother, our minds scream out. The why is evident in street settings. Also, we are designed to pick up patterns in such meaningful settings and learn. You don't even need to teach it.
Fixing learning and education means being able to articulate answers to five questions: why, what, how, when, where we learn. And all our current answers are outdated and one-size-fits-all
There is also the matter of learning design. Math we use is highly compact because of its efficiency. But these can be hostile starting out. There are ways to explore mathematical concepts without using mathematical symbols.
Resonant learning requires the interplay of building competence (how) and comprehension (why) and this can be done well only in meaningful settings (where)
Also, the most profound revelation I've had about learning is from working on our book. Journey of the Mind. Spent twenty years at the intersection of neuroscience and AI but never really got to "understand" and "learn" how the mind works and learns until diving in and then trying to tell its story in an accessible manner. If you are interested in any of this lease do check it out
>>> Meaningless symbols that need to be manipulated to arrive at some mysterious answers.
I get what you're saying, and agree but with a caveat: Please don't take this away completely. There's always going to be a few of us freaks who came to math because of this feature: Math as an exercise in pure abstraction. We have no other refuge. As I've mentioned in this thread and others, proofs were what made math come alive for me. And I didn't struggle with applied math at all.
Agree! And that's because there are some who just get this and are able to explore this in the space of ideas the same way we all have very different tastes in music and literature. Math has both practicality and poetry and many problems are introduced when the focus is one on. What one responds to is personal.
Abstraction can be learned with purpose, eg if they want to program physical movement in a game, X and y need to be variables, not because we are just using abstraction for the fun of it. Even proofs can have meaning and purpose if you can show how they are useful. Algebra, calculus, liberal algebra at least, all make more sense (and easier to accept by new learners) when you actually use them to do something.
Comedian Bengt Washburn describes his school-learning experiences of high-school geometry: "Hours of study. Straight 'A's. And now we are a comedian who can calculate the volume of a cylinder. Height times Pi R squared. You think that comes in handy?"
I'm curious on this. My gut would be that NOTHING transfers between contexts by default. Instead, learning to transfer something between contexts is itself a skill that needs effort.
As an easy example, just counting beats is clearly just counting. Yet counting beats and aligning transitions/changes on or off a beat is a skill you have to work on.
Indeed, the same counting can help people that are running to start to even breaths out. Simple meditations often have you count a breath in, and then count it out. Just because you can count doesn't mean you will automatically be good at any of these things.
There is also the question of what it means to count? Do you literally hear a voice in your head speaking the numbers away? Do you see a ticker? Do you have some other mental tally system?
> My gut would be that NOTHING transfers between contexts by default.
Once you’ve learned to write, you can write with a pen between your toes (crudely because of a lack of fine motor control), with a chisel in wood, with a spray can, men can write while peeing into snow, etc.
> As an easy example, just counting beats is clearly just counting.
Agreed.
> Yet counting beats and aligning transitions/changes on or off a beat is a skill you have to work on.
But that’s not just counting, is it? It’s counting _and_ aligning transitions/changes on or off a beat, so it requires detecting transitions/changes while counting.
The idea of transferring knowledge being a separate skill reminds me of the Wason selection task[1]. I first learned about this in a course on education and it felt pretty shocking to see so many classmates struggling with the logic puzzle version of the question. But then if you set up the same task with a story about being a bartender then it becomes more straightforward to solve.
I'm not clear that writing with different tools is really the same as writing in different contexts? Writing in a different context would be that you have learned to write your letters, but now you are learning to write poetry. Different kinds of poetry, even.
My point for aligning changes on beats is that you are learning to count a mixed radix, effectively. We don't teach it that way, anymore, as positional numbers have grown to be what many of us think of as numbers. But mixed radix counting is the norm in the world in ways that people just don't realize anymore.
> Once you’ve learned to write, you can write with a pen between your toes (crudely because of a lack of fine motor control), with a chisel in wood, with a spray can, men can write while peeing into snow, etc.
This is not accurate. Find a child who has just learned how to write an A, and ask them to write an A with their feet, or even their non-dominant hand. It will be just as hard as getting them to write a B. The connection between shape and motion is a relatively simple one, but your first attempt at writing a word with piss in snow is gonna look awful. Penmanship needs to be learned in the new context.
A fun example I like to bring up involves yelling in foreign languages. Even if you have an impeccable accent in your second language, if you've never practiced talking loudly, the first time you need to order lunch over the noise of a passing subway train, your accent will entirely fall apart as you try to say the same thing, but louder. (Yes, this is a personal anecdote, with a passable accent in my second language, as opposed to impeccable.)
What you describe is one of the fundamental “problems” of associative memories. Which is doing or recalling a thing in one context does not mean you are capable of doing or recalling the exact same thing in another context. Neurons light up based on all the current inputs, and if none of the current inputs light up the neurons that can trigger a skill, good luck doing that skill. This is why practicing in a wide variety of contexts is really important for mastery, you’re essentially increasing the odds that different inputs have a chance to trigger the knowledge that’s locked away in the structure.
Anecdotally, this is true even as an adult. As a non-trad student the application side of things as they are taught are fairly effete, we learn how to translate graphs in algebra, quadratics, polynomials... I don't recall much in the way of meaningful application in either trig or algebra and what was there was remembered solely in the context of future examination.
In one hand I would argue there is virtually no incentive for play, or discovery, or superfluous activity with the math due to grading, and in fact it's disincentivized as it is one factor of a multivariate optimization problem. On the other hand I would argue that it's taught too fast as an effect of the former condition - as someone who isn't in a highly mathematical branch of STEM the use of mathematics is comparably infrequent when considering the TEM, as such atrophy sets rapidly after examination. And this could be said more generally with the S as well, though there is some degree of reinforcement there.
As things are, I feel that the timeline is askew, the won't if these institutions to produce biologists along the same timeline as they did a few decades ago is a little ridiculous considering the ballooning of quantitative discovery that has occured since, for instance, it wasn't so long ago that DNA was a conceptual exercise.
Moreover, the failure of education to keep with the times and adapt a realistic curricula for the modern era is also inhibitory. Indeed I would argue that the current academic zeitgeist is working against itself. At once being a trade program and while also trying to facilitate the development of "academia" itself are forces acting against one another. The number of premed students running the gauntlet in my program far outweigh the number of people with [let's say] legitimate interest in learning about the concepts in the program, which are also made to compete with the premeds in the limited slots available for lab internships. In my experience this leads to a chilling effect. Fortuitously, once in a lab things tend to be a little more facorable in terms of rapport.
My pet theory is that mathematics is largely taught by people who enjoy math, and who see math as a puzzle game with rules you follow to solve puzzles. Kind of like people who are addicted to sudoku or wordle.
Therefore it is taught from this totally abstract perspective and just hooks others who like the game of math. Whereas I think math would have a much greater impact if it was taught from an engineering or science perspective, where math is a tool used to explore the world rather than as something that would be a game you find on the back page of the newspaper.
> My pet theory is that mathematics is largely taught by people who enjoy math, and who see math as a puzzle game with rules you follow to solve puzzles. Kind of like people who are addicted to sudoku or wordle.
> My pet theory is that mathematics is largely taught by people who enjoy math
In most of america, until 5th grade, one teacher teaches everything.
In some states, in 4th grade and 5th grade, you do get some segments of specialization in course load, but even then you still have mostly one teacher.
A lot of applied math courses are also taught by people that despise math and just teach that course because it was given to them and they would much rather teach a course in their main discipline.
Somewhat related but very different from the point you raised, a lot of people that enjoy math, especially research, HATE teaching math. I'd actually even venture it is the majority...
People like John Conway were rare. RIP. I miss seeing him around Fine Hall. And now I think of Nash :(
children wise, i'm thinking like grade 3 where i just had the one teacher for basically everything
the math was really taught by the S&P workbooks. word problems in french that you have to turn into algebra and then solve, and then write out a french answer in words for the problem.
its possible that doung french immersion school is better than just the second language aspects, but i imagine there is/was similar english workbooks
Completely unsurprising. (Former?) Math nerd and common core hater here; having watched myself and other kids go through all of this; beyond EARLY algebra, maybe earlier we literally should not teach any math that doesn't have an immediate and obvious use to children.
(My biggest pet peeve is how Common Core teaches fun -- but unnecessary for learning -- math nerd tricks. I LOVE math nerd tricks, but they should be discovered independently and entirely optional)
Now, the silver lining here is; we have a thing that does hit a lot of advanced high-school math easily. Just let them kids learn video game programming and be done with it.
Do something crazy and teach kids to write their own QP solver. That will force them to learn quadratic functions, vector norms, solving linear equations using LU decomposition, dot products, vector matrix and matrix vector multiplication, convexity, Lagrangian multipliers, etc.
The quadratic programming solver can be used to calculate ordinary least squares for linear regression and controlling robot arms in real time or linear model predictive control.
I would say data structure and algorithms are not more practical to actual literal children. Better off letting them cook/bake, where they have to work out ratios, temperatures, giving one third more or quartering other portions, etc etc. let them do scoring, figure out how to split 300g of chocolate between 5 friends and the like.
Another commenter mentioned that a longer study window would have been more meaningful. I agree, but for very different reasons.
Instead of making this all about "intelligence," I'd argue that, even when something is understood deeply, most people (both kids and adults) are very slow and unsteady when they think about how to apply what they know in a novel context. In other words: you can understand something deeply and fluently and STILL not have this second skill set for transfering between contexts.
So, in summary, if you want to study how able people are at transferring skills, I think you need to lengthen your study window and give people both a lot more time and space to explore and adapt to the new context.
Also, I think the ability to transfer your skills between contexts _is a separate skill_ that you need to learn, in large part, through experience. However, both work experiences and school experiences tend to stress siloing topics and skill sets to make 'test acers' and 'experts,' so very few people get to experience building a deep skill set and then transferring it to a new context early in life, if ever. As a result, honestly, I think almost everyone has a brittle understanding of almost everything they know. The exceptions are outliers. And, when you watch people (or yourself) try to do something new, usually success is, first order, a matter of how comfortable you feel trying things, failing for a bit, and slowly learning how what you know applies to the new context.
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Source: mostly first person experience. I've been teaching math for about half of my life: ~18 of ~36 years, and I still love it. :)
Also, I didn't read through all of the comments, so apologies if someone else already contributed this idea!
My second daughter (16 years old) can bake (including e.g. making a 50% larger amount than the recipe calls for). She can beat me at games where reasoning about probabilities and numbers is involved. She can relate exactly none of that to what she is learning in her algebra classes.
On the surface it's pretty obvious that classroom learning doesn't immediately translate to real world experience, but the paper's finding seems to be more about the extreme degree of discrepancy between the two cohorts. It almost feels like a comment on social class distinctions - there are children who get classroom educations that don't have to use it, while others have to use the skills but don't get the related education.
I'm surprised this was published now, given that I saw a talk on this at a math conference in either 2008 or 2009. The memorable anecdote was that they filmed children who worked with cash in the market, brought them to the classroom, and given pen and paper in the classroom they would be unable to duplicate the calculations they had already done in the market. The speaker was promoting VR to simulate the market context in the classroom.
I guess what this paper adds is a higher N and the reverse case, that classroom skills don't transfer to the market.
When I was an adjunct a couple years ago, I would use money for the more introductory stuff in Python.
I figured that for better or worse, every single person in that classroom will have to deal with some amount of "money math" in their life, and "money math" is still "real" math, and programs involving money are still "real" programs. If nothing else, I couldn't really get the "when will I ever use this????" kinds of questions.
A lot of people seem to have almost a "phobia" of mathematics; they are perfectly fine doing the relevant calculations in regards to stuff that's directly used, like money, but seem to shut down when mathematical notation is used.
Yeah, similarly, a friend of mine's kid got really into Kerbal Space Program. That friend didn't mind his kid playing that one for long periods of time, because there's a ton of real math and physics being used, but the game is relatively fun.
This is really interesting, but I don't agree with this conclusion: "These findings highlight the importance of educational curricula that bridge the gap between intuitive and formal maths." (My own opinion is that educational curricula are generally not very important at all; that people are learning machines that learn what they need to in the contexts they find themselves; and that people -- as shown by this study -- struggle to effectively apply what they've learnt in one context into a different context.)
> My own opinion is that educational curricula are generally not very important at all
We spend an incredible amount of time and effort on educational curricula, so it's worth thinking about.
My opinion is everything you learn before you start actually using knowledge is "just" familiarization. In my opinion, pedagogical instruction should do a much better job of explaining this and incorporating this realization. I do think individual teachers understand this.
This really has nothing to do with pedagogy and everything to do with the difference between calculations and mathematics. It's similar to the split between science and engineering.
In engineering, you're applying rules of thumb to get some desired outcome. Those rules of thumb may be based on math or science, but they could also be practices that have worked before. You don't have to understand why the rules of thumb work to apply them, although it is helpful, if you do.
What those kids are doing is something similar in the world of math. They have a desired outcome -- getting the correct answer, and they have a memorized list of "tricks" that get them to the correct answer quickly. They may or may not know why they work, only that they do.
None of that is really helpful for learning math in a school setting, which is not just about getting the correct answer, but about understanding why the answer is correct, and understanding why those various rules work or don't work. They don't usually even teach mental math tricks in schools because the point isn't being able to calculate quickly. Probably having a catalog of addition tricks makes it even harder to learn basic school arithmetic because you have to ignore what you already know to work back up from first principles.
I can't find it now, but there was a blog article from maybe 15-20 years ago where the author, a math educator, lamented how students seem to put math in a "math box" and completely divorce it from reality.
One example given was a word-problem in which a round-trip was involved and the student complained that there was no way to know they had to double the time, but then demonstrated obvious knowledge of how this worked when asked a real-world question.
Notably, the advanced mathematics guidance programs from the institutional outreach for children were usually by invitation only. Thus, unless you were identified as "gifted" early on, the academic content for your development is very different.
Personally, I think grade-school kids in grade 7 should be introduced to physics with calculus, discreet mathematics, and linear algebra. However, the content should not be part of a graded curriculum, but rather a set of trivial daily puzzles to solve.
The current academic institutions are often engaged in all sorts of policies that have nothing to do with science, or students cognitive development.
Every culture develops 'marketplace' intuitive mathematics; everyone has to make change or figure out how to divide a dozen apples 4 ways.
Only a few cultures develop abstract mathematics. How many have? It's even possible, like the alphabet, that only one culture invented it and others copied that one culture.
HN members may be in a bubble here. In my experience, most people, even those with college degrees (and therefore some exposure to it) have a hard time grasping abstract theory of anything, especially its usefulness, and a harder time applying it. If I say, 'There is no perfect security; we need to make the attack more expensive for the attacker than it's worth to them. The existing system A is wasting resources because it is not protecting something valuable ...'; they ignore everything and might even say 'why are you telling me that?', and then 'will system B perfectly secure this asset?'
It's always alarming for me. How can our world funtion well if all the value of abstract theory is discarded?
Snagged this one recently and his hypothesis seems to jive with these findings.
He suggests that really doing vs math is a largely intuitive process that is then fact checked by logic later on. School, in his view, tends to do the logic side and ignores the intuitive which leads a lot of people to think they’re bad at math especially since they don’t get to practice the intuitive side that many people feel is reserved for those who are just good at math.
Real great read tho. I’ve been working my on intuitive math since then and have found it to be a great experience.
Arithmetic in particular is from India. Indians have a long cultural history of using arithmetic. Even today, my mom who is categorically not a technical person, will randomly use a computational technique I'd never seen before to do her sewing, cooking or measuring. My grandmother, barely educated, did the same thing. They were extremely good at daily calculations. To the point where whatever I picked up off of them helped me win several 'fast math' competitions as a kid in LA. I'm not particularly good at arithmetic.
My guess is that even a not really educated / barely literate Indian kid will be able to do arithmetic, simply due to cultural knowledge.
The schoolchildren do well because they've been exposed to logic as well as arithmetic, and also given a systematic overview of the whole thing, whereas the market kid is just applying whatever was passed down culturally.
However, I don't think we can necessarily transfer this study outside of India to assume that kids from other areas who are forced to work in markets will be good at arithmetic either. In my opinion, this is a unique cultural condition in India which reflects arithmetic's development there. Indian numerals and arithmetical algorithms were first introduced to Europe in the 13th century. The first records of their use in India go back to the first century AD. Thus, it's had a lot more time to 'soak in' to the common culture. I think it would be a good exercise in 'ethnomathematics' (is that a thing) to document all the algorithms used by the various laborers / uneducated productive workers in India. There'd probably be a few gems there.
"ask whether, in the urban Indian context, the arithmetic skills that are used in market transactions transfer to the more abstract maths skills taught in school."
I have a hard time with this notion of 2 different maths. I wonder if it is specific to the "urban Indian context" as the authors seem to suggest in their literature review -- I didn't pursue their references. Intuitive math that is not associated with memorization sounds like g.
My father, who had a PhD in history, had exactly the kind of math skills the study describes in Indian children. With ordinary arithmetic, he was fast and accurate. Money and baseball statistics were no problem. Algebra? No fucking way. As soon as x and y were involved, he struggled. Strike that: he wasn't able to struggle. He wasn't able to engage the gears that would allow him to apply effort.
One time, when I was in high school and already contemplating majoring in math in college, he told me that a math professor had told him that in modern mathematics you didn't have to know what you were talking about. All their theories could apply to anything. Like you could pick up a paper, and they're talking about X, and you could decide X was Donald Duck! He told me this like it was an exotic glimpse into another culture -- he knew that it looked ridiculous to him, but he also knew that it probably looked ridiculous because he didn't understand what he was looking at. You could tell that one part of his brain felt like it was a gotcha moment for the mathematicians, but another part of his brain could see that they weren't embarrassed about it, and he taught WWII every year so he knew that Donald Duck could also be artillery shells or atoms. He had that last defense of common sense that stops people from embracing crank theories about other fields of study.
This was a guy who taught recent history and accepted the abstract ideological struggles of the 20th century without blinking, but when you told him that someone could write an entire doctoral thesis about X without knowing concretely what X was, it was such an alien idea that it was out of range of his curiosity.
From this I would be confused about how he got through high school math, except that my sister, who also has a PhD in history, explained how she got through calculus: she studied all the homework problems and all the solutions over and over until she had memorized them, and she reproduced the patterns on the tests. At our high school, that was good enough for As. In college, it was good enough for Bs.
(It makes me feel a tiny bit more empathy for the condescending mathphobes who denigrate virtually all school mathematics work as pointless, deadening rote learning. For many of them it might be a sincere belief. They might have actually experienced it that way and never experienced any of the worthwhile aspects of it. But, on the other hand, they should have the grace my dad did to stop short of declaring it worthless just because they didn't get it.)
It is very sad to me that so many people can't enjoy that aspect of math. I was lucky, pbs used to show math stuff to kids, so it was fun and interesting before it was a school thing. Of course a huge part of math learning is just hatd work for most if us. But kids should taste the delight first, it motivates them to do the less delightful practice.
Unfortunately, the pleasure or displeasure of doing math compounds quickly. A kid who doesn't enjoy it is going to do the minimum required, and if that isn't enough, it will become ever harder and less enjoyable in the future. You need practice, and practice is a dirty word in pedagogy.
Someday education is going to catch up with music and sport in its attitude towards practice. As adults we all understand that in music or sports, the most elite of the elite, the top 0.01% of human beings in attainment, are never too good to need more practice and polish of basic skills. But when we look at children learning math, repetition becomes anathema. My question is, how could anybody ever enjoy math without repetition? You need to make the boring stuff easy and then keep it easy. How does that happen without repetition? If you don't practice, the boring stuff becomes hard again, and you don't have brains to spare for the interesting stuff.
I can’t relate to this at all. In music, you practice until you get it, and then you stop. It’s important not to over-practice, in fact. In high school math classes, I was assigned an order of magnitude more practice problems than I actually needed. There was no “practice until you get it, then quit.” It was “do all the problems whether you understand them or not, or you get a bad grade.” Repetition is absolutely anathema when you’ve ceased to learn anything from it.
I agree. Kids need to eat their vegetables, but we can make vegetables quite delicious if we try. Also, the music metaphor is imperfect because some musicians are performers others composers. I prefer sport as the metaphor: a mix of short laregely repituous training and longer term strategy, different styles for different athletes, influence of genetics and talent heterogeneity acknowledged at the elite end of the spectrum.
Very interesting study. I guess even without conscious rote learning there's repetition involved in selling the same set of vegetables in similar quantities over months, years and customers paying in similar bills - after all there is only a few of each. So it is not surprising those skills are not transferable.
However, it does show the kids are not "dumb" and schools need to make formal maths more interesting. More than the curriculum itself, teachers can make a difference, if well trained.
Anecdotally I do find common maths skills have a strong correlation to general competence as perceived by others. I have always been impressed by executives that have firm grasp of numbers, even in non-financial jobs and I find they do well in their careers. They can instantly notice something that stands out, smells bad etc.
A store clerk that pulls out a calculator to do simple maths gives out bad vibes.
> A store clerk that pulls out a calculator to do simple maths gives out bad vibes.
Conversely, store clerks in corner stores that use print roll tabulating calculators to add everything and reach totals give confidence to many, particulalrly those that hand over the printouts for inspection.
There are patterns of argument that serial scammers use to confuse clerks about change.
One of the better defence strategies is to slow walk the asserted calculation through on a calculater with print tally.
It means almost everything is a skill that needs to be learned. Thus being successful in a school setting could mean one has successfully learned how to succeed at exams, but not really learning anything useful. I recently needed to take a test for a license of some kind, and while preparing with a set of questions noticed I can pick the correct answer without even reading. The correct answer was always much longer than the wrong answers and much more formal which was obvious by even just skimming.
It also means the differences between people and their potential are in general for the vast majority of people small. Truly gifted or special is rare, as is truly giftless.
Jean Piaget was one of the foundational researchers in cognitive development, particularly in 'constructivist' circles. Constructivism is the theory that learning is an active psychological process, and is contrasted with behaviourism (often associated with B.F. Skinner), in which learning is a passive process. Constructivism is popular amongst school teachers and behaviourism is unpopular.
Piaget's theories are very much rooted in learning about the physical world, and are thus more popular amongst teachers in the 'STEM' disciplines. The other foundational researcher in constructivism was Lev Vygostky, who worked in the Soviet Union under Stalin, and his theories reflect the political pressures of that environment; his earlier work remains influential especially amongst humanities teachers, whereas his later work is excessively ideological.
Piaget proposed a stage-based model of development, in which children progress through four different stages of cognition. The last two stages he called the concrete operational stage and the formal operational stage. In the concrete operational stage, children become capable of one layer of abstraction or symbolic indirection; it is not until they reach the formal operational stage that they are capable of multiple layers of abstraction or symbolic indirection.
It is interesting to note that Piaget suggested that the transition between these two stages usually occurs around the age of eleven, which is when British children transition from primary to secondary school.
So, to contextualise the linked article, the studied children developed an applied understanding of mathematics that fit their concrete operational cognition: the mental operations were readily made concrete by manipulating coins and banknotes.
However, when children study mathematics in school, it is very much done as formal operations: the numbers are not intended to represent anything physical at all. Consequently, many children learn mathematics as a set of arcane rituals for manipulating symbols on paper, because they can't yet understand the abstract meanings of those symbols.
"many children learn mathematics as a set of arcane rituals for manipulating symbols on paper, because they can't yet understand the abstract meanings of those symbols."
In the US?
I had pretty typical US public school education. Word problems and application were ubiquituous. Perhaps my experience is non-representative, or you like the study authors are speaking of other educational contexts (Asia).
I grew up in Australia and taught in Australia and the UK; and yes, the word problems are ubiquitous in those countries too. But those word problems are always deeply inauthentic, and extracting the relevant information from them becomes just another arcane activity.
One that comes to mind involves a farmer with a given length of fencing, and the student has to find the area of the largest rectangular field the farmer can surround with that fencing. It's a good mathematical puzzle, but the actual real-world problem is how much fencing the farmer needs to surround a given field.
Coming up with genuine, real-world applications for every mathematics lesson is extremely time-consuming, and maths teachers simply don't have the time.
With ChatGPT et al... it should start to be pretty easy. I envision a question answer game between humans and the AI. The AI would set up the scenario and the kids would have to ask the right questions. Teachers could supervise and evaluate
Maybe there's an element of both time and skill? How many folks that pursue math can point to a great early teacher as being influential. Not all, but I want to believe it's common and maybe true for most.
Some children read word problems, understand the question and apply math. Most children look for key words, use those key words to guess what operation to apply, then apply it to the numbers in the question (e.g. there were 3 numbers and the word "total" so I'll sum the numbers).
It's pretty hard to believe the article's authors when the third sentence in the abstract labels "arithmetic" as "complex calculations". There is nothing complex about arithmetic.
Further evidence that human children are just LLMs whose 'capabilities' do not generalize outside the training data or are robust to even minor variation of the abstract algorithm being tested. Very sad.
Even as a maths professional (20 years ago), I had trouble programming the same abstract algorithms to be fast and generally usable. It is ... non-intuitive for me.
That is why I always admired work of people like Peter Montgomery or Donald Knuth.
Agreed. Strong abstract math skills often signal cognitive strength and discipline, predicting success in complex real-world tasks more than direct transfer of content.
How would you take into account the converse finding by TFA?
> By contrast, children with no market-selling experience (n = 471), enrolled in nearby schools, showed the opposite pattern. These children performed more accurately on simple abstract problems, but only 1% could correctly answer an applied market maths problem that more than one third of working children solved (β = 0.35, s.e.m. = 0.03; 95% confidence interval = 0.30–0.40, P < 0.001). School children used highly inefficient written calculations, could not combine different operations and arrived at answers too slowly to be useful in real-life or in higher maths.
This relates to the (somewhat controversial) economic theory that much of higher education is not about training or creating transferable skills, and instead is about "signaling". Essentially, we place children in simulated quasi-work settings and use it to determine who has the most of certain economically valuable abilities. By performing well at abstract mathematics, a child signals that they have the requisite mental ability to tackle similarly complex "real world" problems. But also, they signal that they are capable of obeying authority, working with diligence, delaying gratification, etc.
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