But a math class is a large set of items. It's akin to giving a bunch of very hard matrix items and defining passing as getting most of them right. Clearly those who pass will have higher IQs than those who do not. In other words maybe someone with an 85 IQ can solve a single integral with enough time and effort, but they'll never pass the AP calculus exam. I conclude they do not understand calculus sufficiently in this case. Now it would be nice to know the average difficulty of a calculus exam question.
There are ways to get across math concepts even to lower-intelligence people, I taught the basics of complex numbers, trig. and calculus to ~130 IQ 10 year-olds in passing while teaching electronics. You can integrate the flow of a dripping faucet with a bucket, or the area of a figure by cutting it out and weighing it. Mathematicians are always putting the obvious in terms of the incomprehensible. For instance the delta-epsilon formulation of the mean value theorem just says that if a continuous line goes into a box and comes out, it went through. Babies could understand it. I have a top 1 in several thousand IQ, independently wrote a paper on null subspaces of Minkowski spaces at 12, and it took me 20 minutes of puzzling over the obtuse phrasing in my calculus textbook to figure what it said.
Vocabulary and analogies are more irreducibly g-loaded.
On the Woodcock-Johnson and Stanford-Binet individually-administered IQ tests published by Riverside, all questions have close to the same discrimination parameter, and scores are calulated as if it were identical for all questions. This surprised me, but it is necessary if score calculations are to be tractable. The discrimination is quite low: ability 1 s.d. below the question difficulty gives a 25% chance of being correct, if ability equals difficulty, 50%, and if ability is 1 s.d. above difficulty then 75%.
I highly recommend studying the WJ-IV technical manual linked in my post on practical implications of real, not age-normed measures of intelligence, particularly the section on W-score scale construction: https://substack.com/@enonh/p-149185059
>Vocabulary and analogies are more irreducibly g-loaded.
Lol any 95 iq immigrant can learn a new language. They will never learn calculus. Apparently something is wrong with your tests, to get such an absurd result. After all, like language, classic IQ tests are absurdly trainable, see flynn effect etc.
The problem is that math is also trainable, although to a lesser extent than words. So you need to compare people with similar time spent learning. High schools provide a natural experiment to exploit for this purpose. It appears that the mean SAT of a 5 scorer on AP calculus for instance is in the 1400s somewhere which should translate to 120-something IQ, as I would predict.
Mathematicians have lower scores than philosophy or even economics professors. Typically professors have just around 115IQ anymore, used to be low 120s with an 11-12 pt. s.d. on the WAIS in the ‘50s at Cambridge (pre-Flynn effect, lower in today's terms) with the top score of around 140 in a sample of a few dozen. The math profs were pretty average, the physicists and biochemists were a bit brighter. That the Flynn effect does not affect verbal but does affect non-verbal reflects the higher g-loading of the former. The WJ manual I referred to has quite a bit on the factor structure and loadings of the many, many subtests.
Math, as math teachers do it, is empty symbol shuffling. As truly intelligent people do it, it's visualization and understanding meaning and purpose of what one is visualizing. Vocabulary, to a “packer” mentality who believes that learning is the stacking up of knowledge packets, means learning a word's meaning in terms of other words. To a “mapper”, each vocabulary word maps to a concept in an integrated map of reality, each giving a distinct mode of thought that expands what can be thought about. The mapper mind is a higher order of being than the packer, the difference between genius and bookkeeper. Search for Alan G. Carter's Reciprocality on the mapper/packer divide and how to cultivate mapping thought.
Then why is a high GRE Verbal score out of reach for a typical monolingual 100 IQ woman seeking a master's in education? Can't she just brute-force a bunch of flash cards?
Arthur Jensen explained well why vocabulary is the most g-loaded measure of crystallized intelligence: a word fills a conceptual slot, and people vary far more in their depth and breadth of slots than they do in their vocabulary exposure. Words with the best discrimination indices aren't obscure jargon that can easily be depicted concretely; they're words with specific abstract meanings that mustn't be overfit or underfit. If someone can't grok the conceptual niche that a word is filling, the word just can't stick, at least not such that it'd be used correctly.
Math is also words, I consider it to be a subset of natural language with a bunch of nice properties. It would seem the threshold for understanding the conceptual slots of math is higher than for individual natural words. For reference, I maxed out the GRE V and find a lot of people who can also do that who can't grasp the math I do. So I consider the math to be of a higher nature.
There are plenty of people who score better on SAT/GRE Math than on the verbal section. Not everyone who even maxes out GRE-M maxes out verbal too. The verbal portion of the SAT at least is known to be notoriously harder to improve through studying.
The g factor split into its two main factors, correlated but separable, is not "verbal vs. mathematical" but "verbal vs. spatial." Math draws upon both verbal abilities (for reasons you just described) and spatial abilities (especially for the sort of math relevant in the sciences and engineering). There are cognitive tasks that *must* be mapped onto a quasi-spatial structure, and there are cognitive tasks that mustn't (can't) be - and non-spatial cognitive resources only go so far for spatial problems, and vice versa. Yes, even math and programming concepts that you'd think are too abstract to be visualized depend on mastering prereq math that is clearly spatial; they sit upon learned scaffolds that you've compressed and automatized to the point that they seem invisible to you.
It's been argued that there's some >99th percentile spatial talent that goes unrecognized when we only assess verbal and math performance directly, even though the spatial component has the best predictive power in some technical domains: https://cdn.vanderbilt.edu/t2-my/my-prd/wp-content/uploads/sites/826/2013/02/Lubinski_2010_spatial.pdf Similarly, some >99th percentile non-spatial abstraction ability wouldn't be captured by math performance - such performance is a compound measure constrained by spatial ability in a way that performance in non-spatial domains is not.
So sure, "any 95 IQ immigrant" can learn a language for functional purposes, just like she can surely learn algebra. She can't perform high-level verbal abstraction within it. She can't necessarily compress many superficially different instances into a single abstract category, spot implicit constraints and unstated assumptions in a text, organize her internal vocabulary into proper semantic webs rather than a disorganized list, or spontaneously come up with clarifying analogies. These are real cognitive powers based on non-spatial abstraction ability, and to the extent that tests can measure them, they're not easily trainable.
I can accept a certain amount of controversial and radical takes on IQ when backed by sufficient empirical evidence rather than speculative interpretations and logical chains that make a casual hypothetical or deduced argument about IQ generalizations. But this thread is simply ridiculous and gross… Overly culturally biased and American-centric reasoning that clearly doesn’t apply to many countries, especially East Asian countries, where there are no lazy ass at school too pervasive to assume that unintelligent = lazy, given their highly competitive and challenging education system. Math is not a sufficient proxy for g, and academic achievement in math depends on people as well as on how good a country is at teaching math. Math can be a good proxy for IQ when mathematical teaching is very poor, like in the US, where only smart ass can’t fall behind with poor teaching, but it becomes a significantly weaker predictor when mathematical teaching gets much better (or at least is extremely overvalued culturally or with higher academic standards to raise the math level of lower IQ individuals with just better intuitive teaching of complex topics and hustle culture in schools, so it’s very culturally dependent to make qualitative assumptions on IQ range in the general population, even with a highly g-loaded domain like math), such that it’s very easy for low IQ from less developed countries to outsmart high average IQ Americans, and so it can be less discriminant for intelligence… And it’s not because we don’t have much qualitative research on IQ that these poor and speculative arguments can be justified by it.
It's neither ridiculous nor gross. I cite data that provides some evidence for my hypotheses. It's also not "gross", homosexuality and polyamory are examples of gross things, intelligence research is clearly not like these.
Sorry my comment was sent too early to develop my points. But yet your cited data are too little to provide evidence for your hypotheses compared to the overwhelming data that can contradict them very easily and you misunderstood the meaning of gross I had (fun fact : gross is polysemic)
teacher quality is known not to really matter. as for asian strivers, yeah if you spend quadruple the time on it maybe you memorize a few more techniques, but what is really interesting is how their genetic intelligence predicts their understanding given they are taught just like everyone else. Also I've found grinder asians usually don't understand math, they just memorize problem techniques which is similar to working with your hands. that's why they're obsessed with math competitions instead of doing scientific modeling or more productive deep pursuits with math
I think you're over-estimating the level of math needed in every day life. Most jobs don't even need high school algebra. Even jobs that deal with spreadsheets and numbers like insurance agents only need to know basic arithmetic.
A bigger problem is in understanding probability and statistics as it relates to news stories. Stories about crime, health outcomes, etc do require some understanding of statistics that are beyond the grasp of people with less than 115 IQ. But these topics tend to be very politicized so for people with average IQs their CONFIRMATION BIAS takes hold. They actually filter out anything they don't already agree with. So this is a moot point anyways.
Even for people at +2 SD (130+) who can gather, organize , and model/interpret data, they have a hard time overcoming confirmation bias as well.
There is a paper called "Nonlinear Psychometric Thresholds for Physics and Mathematics" which I think applies here. It shows that you likely need an IQ of at least 115 to have a good shot at achieving a GPA of 3.5 or higher in math or physics. I also remember some recent estimates showing math graduates have an IQ of about 115 nowadays? I think that your category of IQ 130+ needed to achieve a B/C grade in undergrad maths is more realistically around 115.
Your model treats intelligence as efficiency within a fixed problem space. That’s useful, but it’s not the whole thing. IQ is good at measuring how well someone moves from A to B when the route and rules are already defined. What it struggles to capture is a different cognitive move: noticing when the route itself is wrong.
Some people optimize known paths. Others pause, explore, and test odd possibilities that look like distraction or laziness under standard metrics. Most of the time, that exploration goes nowhere. Occasionally, it reveals a shortcut that reframes the entire problem. Once that shortcut is formalized, high-IQ optimizers adopt it and it looks obvious in hindsight. Our measurements are excellent at grading performance inside a model, and poor at detecting who will break or replace the model. That doesn’t make IQ meaningless, but it does make it incomplete. The mistake isn’t measuring intelligence. It’s mistaking model-optimization for intelligence itself.
There are no thresholds in real life. Or in item response theory terms, no item has infinite discrimination (loading =1).
But a math class is a large set of items. It's akin to giving a bunch of very hard matrix items and defining passing as getting most of them right. Clearly those who pass will have higher IQs than those who do not. In other words maybe someone with an 85 IQ can solve a single integral with enough time and effort, but they'll never pass the AP calculus exam. I conclude they do not understand calculus sufficiently in this case. Now it would be nice to know the average difficulty of a calculus exam question.
There are ways to get across math concepts even to lower-intelligence people, I taught the basics of complex numbers, trig. and calculus to ~130 IQ 10 year-olds in passing while teaching electronics. You can integrate the flow of a dripping faucet with a bucket, or the area of a figure by cutting it out and weighing it. Mathematicians are always putting the obvious in terms of the incomprehensible. For instance the delta-epsilon formulation of the mean value theorem just says that if a continuous line goes into a box and comes out, it went through. Babies could understand it. I have a top 1 in several thousand IQ, independently wrote a paper on null subspaces of Minkowski spaces at 12, and it took me 20 minutes of puzzling over the obtuse phrasing in my calculus textbook to figure what it said.
Vocabulary and analogies are more irreducibly g-loaded.
On the Woodcock-Johnson and Stanford-Binet individually-administered IQ tests published by Riverside, all questions have close to the same discrimination parameter, and scores are calulated as if it were identical for all questions. This surprised me, but it is necessary if score calculations are to be tractable. The discrimination is quite low: ability 1 s.d. below the question difficulty gives a 25% chance of being correct, if ability equals difficulty, 50%, and if ability is 1 s.d. above difficulty then 75%.
I highly recommend studying the WJ-IV technical manual linked in my post on practical implications of real, not age-normed measures of intelligence, particularly the section on W-score scale construction: https://substack.com/@enonh/p-149185059
>Vocabulary and analogies are more irreducibly g-loaded.
Lol any 95 iq immigrant can learn a new language. They will never learn calculus. Apparently something is wrong with your tests, to get such an absurd result. After all, like language, classic IQ tests are absurdly trainable, see flynn effect etc.
The problem is that math is also trainable, although to a lesser extent than words. So you need to compare people with similar time spent learning. High schools provide a natural experiment to exploit for this purpose. It appears that the mean SAT of a 5 scorer on AP calculus for instance is in the 1400s somewhere which should translate to 120-something IQ, as I would predict.
Mathematicians have lower scores than philosophy or even economics professors. Typically professors have just around 115IQ anymore, used to be low 120s with an 11-12 pt. s.d. on the WAIS in the ‘50s at Cambridge (pre-Flynn effect, lower in today's terms) with the top score of around 140 in a sample of a few dozen. The math profs were pretty average, the physicists and biochemists were a bit brighter. That the Flynn effect does not affect verbal but does affect non-verbal reflects the higher g-loading of the former. The WJ manual I referred to has quite a bit on the factor structure and loadings of the many, many subtests.
Math, as math teachers do it, is empty symbol shuffling. As truly intelligent people do it, it's visualization and understanding meaning and purpose of what one is visualizing. Vocabulary, to a “packer” mentality who believes that learning is the stacking up of knowledge packets, means learning a word's meaning in terms of other words. To a “mapper”, each vocabulary word maps to a concept in an integrated map of reality, each giving a distinct mode of thought that expands what can be thought about. The mapper mind is a higher order of being than the packer, the difference between genius and bookkeeper. Search for Alan G. Carter's Reciprocality on the mapper/packer divide and how to cultivate mapping thought.
Then why is a high GRE Verbal score out of reach for a typical monolingual 100 IQ woman seeking a master's in education? Can't she just brute-force a bunch of flash cards?
Arthur Jensen explained well why vocabulary is the most g-loaded measure of crystallized intelligence: a word fills a conceptual slot, and people vary far more in their depth and breadth of slots than they do in their vocabulary exposure. Words with the best discrimination indices aren't obscure jargon that can easily be depicted concretely; they're words with specific abstract meanings that mustn't be overfit or underfit. If someone can't grok the conceptual niche that a word is filling, the word just can't stick, at least not such that it'd be used correctly.
Math is also words, I consider it to be a subset of natural language with a bunch of nice properties. It would seem the threshold for understanding the conceptual slots of math is higher than for individual natural words. For reference, I maxed out the GRE V and find a lot of people who can also do that who can't grasp the math I do. So I consider the math to be of a higher nature.
There are plenty of people who score better on SAT/GRE Math than on the verbal section. Not everyone who even maxes out GRE-M maxes out verbal too. The verbal portion of the SAT at least is known to be notoriously harder to improve through studying.
The g factor split into its two main factors, correlated but separable, is not "verbal vs. mathematical" but "verbal vs. spatial." Math draws upon both verbal abilities (for reasons you just described) and spatial abilities (especially for the sort of math relevant in the sciences and engineering). There are cognitive tasks that *must* be mapped onto a quasi-spatial structure, and there are cognitive tasks that mustn't (can't) be - and non-spatial cognitive resources only go so far for spatial problems, and vice versa. Yes, even math and programming concepts that you'd think are too abstract to be visualized depend on mastering prereq math that is clearly spatial; they sit upon learned scaffolds that you've compressed and automatized to the point that they seem invisible to you.
It's been argued that there's some >99th percentile spatial talent that goes unrecognized when we only assess verbal and math performance directly, even though the spatial component has the best predictive power in some technical domains: https://cdn.vanderbilt.edu/t2-my/my-prd/wp-content/uploads/sites/826/2013/02/Lubinski_2010_spatial.pdf Similarly, some >99th percentile non-spatial abstraction ability wouldn't be captured by math performance - such performance is a compound measure constrained by spatial ability in a way that performance in non-spatial domains is not.
So sure, "any 95 IQ immigrant" can learn a language for functional purposes, just like she can surely learn algebra. She can't perform high-level verbal abstraction within it. She can't necessarily compress many superficially different instances into a single abstract category, spot implicit constraints and unstated assumptions in a text, organize her internal vocabulary into proper semantic webs rather than a disorganized list, or spontaneously come up with clarifying analogies. These are real cognitive powers based on non-spatial abstraction ability, and to the extent that tests can measure them, they're not easily trainable.
I can accept a certain amount of controversial and radical takes on IQ when backed by sufficient empirical evidence rather than speculative interpretations and logical chains that make a casual hypothetical or deduced argument about IQ generalizations. But this thread is simply ridiculous and gross… Overly culturally biased and American-centric reasoning that clearly doesn’t apply to many countries, especially East Asian countries, where there are no lazy ass at school too pervasive to assume that unintelligent = lazy, given their highly competitive and challenging education system. Math is not a sufficient proxy for g, and academic achievement in math depends on people as well as on how good a country is at teaching math. Math can be a good proxy for IQ when mathematical teaching is very poor, like in the US, where only smart ass can’t fall behind with poor teaching, but it becomes a significantly weaker predictor when mathematical teaching gets much better (or at least is extremely overvalued culturally or with higher academic standards to raise the math level of lower IQ individuals with just better intuitive teaching of complex topics and hustle culture in schools, so it’s very culturally dependent to make qualitative assumptions on IQ range in the general population, even with a highly g-loaded domain like math), such that it’s very easy for low IQ from less developed countries to outsmart high average IQ Americans, and so it can be less discriminant for intelligence… And it’s not because we don’t have much qualitative research on IQ that these poor and speculative arguments can be justified by it.
It's neither ridiculous nor gross. I cite data that provides some evidence for my hypotheses. It's also not "gross", homosexuality and polyamory are examples of gross things, intelligence research is clearly not like these.
Sorry my comment was sent too early to develop my points. But yet your cited data are too little to provide evidence for your hypotheses compared to the overwhelming data that can contradict them very easily and you misunderstood the meaning of gross I had (fun fact : gross is polysemic)
teacher quality is known not to really matter. as for asian strivers, yeah if you spend quadruple the time on it maybe you memorize a few more techniques, but what is really interesting is how their genetic intelligence predicts their understanding given they are taught just like everyone else. Also I've found grinder asians usually don't understand math, they just memorize problem techniques which is similar to working with your hands. that's why they're obsessed with math competitions instead of doing scientific modeling or more productive deep pursuits with math
I think you're over-estimating the level of math needed in every day life. Most jobs don't even need high school algebra. Even jobs that deal with spreadsheets and numbers like insurance agents only need to know basic arithmetic.
A bigger problem is in understanding probability and statistics as it relates to news stories. Stories about crime, health outcomes, etc do require some understanding of statistics that are beyond the grasp of people with less than 115 IQ. But these topics tend to be very politicized so for people with average IQs their CONFIRMATION BIAS takes hold. They actually filter out anything they don't already agree with. So this is a moot point anyways.
Even for people at +2 SD (130+) who can gather, organize , and model/interpret data, they have a hard time overcoming confirmation bias as well.
I don't recall claiming that calculus is heavily used by most people day to day. Clearly it cannot be, since <15% of people can become skilled at it.
There is a paper called "Nonlinear Psychometric Thresholds for Physics and Mathematics" which I think applies here. It shows that you likely need an IQ of at least 115 to have a good shot at achieving a GPA of 3.5 or higher in math or physics. I also remember some recent estimates showing math graduates have an IQ of about 115 nowadays? I think that your category of IQ 130+ needed to achieve a B/C grade in undergrad maths is more realistically around 115.
Looks like 130 is where you can make mostly As. I was thinking of princeton and not oregon state grading scale at any rate
Your model treats intelligence as efficiency within a fixed problem space. That’s useful, but it’s not the whole thing. IQ is good at measuring how well someone moves from A to B when the route and rules are already defined. What it struggles to capture is a different cognitive move: noticing when the route itself is wrong.
Some people optimize known paths. Others pause, explore, and test odd possibilities that look like distraction or laziness under standard metrics. Most of the time, that exploration goes nowhere. Occasionally, it reveals a shortcut that reframes the entire problem. Once that shortcut is formalized, high-IQ optimizers adopt it and it looks obvious in hindsight. Our measurements are excellent at grading performance inside a model, and poor at detecting who will break or replace the model. That doesn’t make IQ meaningless, but it does make it incomplete. The mistake isn’t measuring intelligence. It’s mistaking model-optimization for intelligence itself.