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…
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.
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.
> 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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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
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.
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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.
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…
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.
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.
1. Smart person makes lots of money
2. Smart person has children
3. Smart person's children are likely to be smarter, and guaranteed to have more money
Sounds about right to me
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.
https://www.wsj.com/articles/colleges-bend-the-rules-for-mor...
https://www.hollywoodreporter.com/lifestyle/lifestyle-news/h...
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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.
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.
Also, one might get the impression from your reasoning that you find that having money makes you smarter.
Dead Comment
This doesn't even come close to being true.
This hints it’s not actually making people more productive. It just lets them get a bigger piece of the pie.
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.
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.
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.
https://v.cx/2010/04/feynman-brazil-education
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.
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.
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.
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.
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.
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!
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.
When I was taught long division I was told how it worked, although it is fairly obvious if you think about it for a few seconds anyway.
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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.
Hint: I might need to google the authors of the study, I don't need to google Richard Feynman...
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.
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.
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)
https://blog.comini.in/p/why-khanmigo-will-fail
https://blog.comini.in/p/what-happens-in-math-class
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
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.
From: "How We Form and Defend Stupid Opinions" https://www.youtube.com/watch?v=9qVNZLLK-yg
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?
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.
[1]: https://en.wikipedia.org/wiki/Wason_selection_task
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.
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.)
My curiosity is that this is my prior. The article is clearly framed in the opposite direction. Am I just putting too much emphasis on the headline?
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 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.
But I agree we could achieve same results by teaching more practical skills. Maybe, programming (data structures and algorithms) over calculus?
That said, spending a bunch of time memorizing stupid identities to compute integrals is probably not the best way to teach introductory calculus.
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.
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.
I wish this was the case.
You mean in college?
I can't imagine how this is true on basic arithmetic level.
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 :(
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