I hope you don't think this article in any way supports your belief that some people can't learn algebra/calculus due to any kind of genetic traits, because I don't think the complexity of either rises to the level of "difficulty" that only gifted minds can comprehend. Very, very little in this world does.
Your belief should probably be considered heretical, even among your conservative colleagues, because your colleagues should recognize how deficiencies in one area can be made up for in other areas, sometimes (but not always) at a higher efficiency cost.
Where genetic gifts are lacking, determination and perseverance can almost always make up the difference, especially at a basic algebra/calculus level.
While I'm sure almost everyone can learn the basics of algebra or calculus, no amount of wishful thinking will give everyone the brain needed to master them or even be proficient. I'm not sure where your belief that perseverance can overcome all even comes from.
I have been trying to get better at chess lately. I grew up playing it so I'm already fairly proficient, and I've been putting a decent amount of free time into studying openings and playing puzzles. Yesterday, I played my friend who hasn't played in literally years and never really played seriously anyway, and he beat me handily 3/3 games. I know that I do not have the brain to be a chess master, no matter how much time I put in, because I don't have enough working memory to keep the board in my head.
I disagree, as it's not a sufficiently difficult topic to warrant some kind of genetic stratification.
Life circumstance stratification, certainly, and you will get no disagreement from me that some people don't have the time or energy necessary to devote to something as not-immediately-useful as calculus, but from an intellect perspective, no.
Everyone who is not experiencing some kind of mental illness has the intellectual capacity to learn calculus, though I will capitulate it is fortunate for me and this statement that we do classify a low enough IQ as a mental illness!
It’s not a sufficiently difficult topic for you. Have you ever taught algebra to a classroom of people from a wide range of intellectual capabilities and experiences? Are you even aware of some of the pain points to learning algebra?
Do I need experience teaching calculus in order to suggest that it's got more to do with environment and situation than it does intellectual capability? No, I don't think that's true.
Are you seriously suggesting trying to raise multiple children while simultaneously learning calculus is approximately as hard as living alone and trying to learn calculus?
Honestly, I think you're trying to find a neat solution where none exists. You haven't stumbled on anything here, you're more likely ignoring the real-life circumstances your students find themselves in, and your colleagues are not.
> Do I need experience teaching calculus in order to suggest that it's got more to do with environment and situation than it does intellectual capability?
Yes, because I'd expect HN readers to understand an ad hominem when they see one.
I could be living in a ditch down by the river, or I could be the leading researcher into how people learn calculus and it wouldn't make a difference with regard to my argument (though, practically speaking if I were the lead researcher on how people learn calculus, I'd be more likely to back my argument up with objective research, and that would strengthen my argument).
Ad hominem occurs when you use someone's reputation in another area to discredit someone on an unrelated matter. For example, you're the worst golfer on the planet so I say you don't know how to cook. Golf expertise has no bearing on your cooking skills. But if you consistently burned toast or you mainly use your oven to heat up frozen pizza, I'm going to take anything you say about baking a cake with a giant grain of salt.
No, ad hominem occurs when you use who a person is rather than their argument to claim they are incorrect. Here's a definition from Google:
> (of an argument or reaction) directed against a person rather than the position they are maintaining.
Nothing about a reputation or using it to discredit them.
For example, I'm not a calculus teacher, therefore I couldn't possibly form a valid argument about how teaching works. That would be an ad hominem, because it focuses on who I am (not a teacher), rather than what I've said (the teacher I was replying to hasn't eliminated any variables at all before drawing their conclusion). (It's also nonsensical, considering how many other things people teach, and how small a percentage of all teachers calculus teaching ends up being, and how unrelated-to-the-science-of-education calculus is).
It's actually funny, because the example you've given isn't an ad hominem, since you have evidence to support the idea that I can't cook (the burnt toast). You're equating an absence of information about me with specific data, which is different.
You don’t know what an ad hominem is. You are incorrectly applying the definition. It is not an argumentative fallacy to ask the basis by which a person’s assertions have been formed. You’ve made lots of claims about teaching but clearly you have no experience to back it up and (this is important) you have not provided any citations to back up your assertions.
People have to make judgments with imperfect knowledge. It’s reasonable to discount the unsubstantiated opinions of someone with no experience with the topic at hand.
It should at least be interesting to you that it was obvious from your comments that you don’t have experience teaching mathematics in the classroom. Why was that so obvious to those of us with that experience? The previous question is rhetorical.
I didn't say it was an argumentative fallacy to ask the basis by which a person's assertions have been formed, I said it was a fallacy to say a person's assertions are wrong because of some aspect of themselves, which is what's taken place here by insisting I must be a calculus teacher in order to challenge your assertion that IQ is the primary source of the problems your students have with learning in your classrooms.
What is interesting to me is the fact that you retreated to this ad hominem the moment you were challenged, because it tells me you don't have any real explanation for how you eliminated other possible causes for your students sometimes performing poorly.
You'd prefer to live in a world where your experience has meaning than to live in a world where your experience is not valuable when faced with this question, which is completely human of you, but ultimately not useful in this discussion, due to its anecdotal and un-rigorously collected nature.
I can't stop you from throwing this New Yorker article, and the other works of Dr. Harden, in the face of your colleagues, but I can hopefully dissuade others from making the same logical mistakes you're making. I believe I've succeeded at that, by clearly highlighting the carelessness of what you've said here.
Ultimately, what I find most fascinating is, in real time, you've demonstrated how right Dr. Turkheimer ultimately is and how dangerous this research can be when put in the hands of folks who don't understand its delicacy or even the basic facts surrounding these arguments.
I'm grateful for your engagement with me, it's been helpful to work through this with someone like you, but I'll commit to the thing you tried and failed to do; I'm no longer going to reply to your comments in this thread. You're clearly (and I mean clearly) wrapped up in a need to think of some of your students as too dumb to learn calculus, and there's literally nothing I or anyone else here can say that would convince you otherwise, and at this point I've done my part in preventing others from thinking that your insight is useful or helpful in this conversation.
I committed to not responding to you in a different thread. I did not commit to not responding to you in all threads. The teacher in me forced me to try to explain to you why your use of ad hominem was incorrect. My previous response had nothing to do with what was being discussed as such. Your conclusions have not been logically valid.
Here’s an example:
… I must be a calculus teacher in order to challenge your assertion that IQ is the primary source of the problems your students have with learning in your classrooms..
No one has said any of these things and no one has implied any of these things. I never said or implied that IQ is the primary source of anything. No one believes that you must be a teacher of calculus to be right. What people have wondered is if you are a teacher because some of your statements seem to the the type of statements only a non teacher would make.
Oh, well, if you're a leading researcher in math pedagogy, you should just say so and save time. Such credibility would offer serious firepower to the debate.
> I have been trying to get better at chess lately. I grew up playing it so I'm already fairly proficient, and I've been putting a decent amount of free time into studying openings and playing puzzles. Yesterday, I played my friend who hasn't played in literally years and never really played seriously anyway, and he beat me handily 3/3 games. I know that I do not have the brain to be a chess master, no matter how much time I put in, because I don't have enough working memory to keep the board in my head.
Given the immense complexity involved with learning, there's essentially no way for you to describe the situation with enough detail for anyone to be able to make an informed judgement. It's extremely common for bad players to describe themselves as 'proficient', spending a 'decent amount of time' can mean anything, studying openings and playing puzzles is likely the wrong way to practice, the skill level of your friend is impossible to ascertain, etc.
The way you actually get better at anything is by:
1. Dedicating enough time to it, consistently (~20h per week or so maybe if you want any sort of quick results). The incredibly common trap here is that people think that spending time on a game equals getting better at it. There's no way to improve without spending a certain amount of time playing, but spending time playing does not make you better on its own.
2. Doing whatever it is you want to get good at (doing anything else does not count - if you want to learn how to play regular chess, play regular chess; puzzles or anything else does not count).
3. Reviewing your games to find mistakes you've made - this part is crucial, if you can't see any mistakes you made, you cannot improve. If that's the case, get someone better than you to review your game(s).
4. Playing while trying to work on addressing one mistake at a time, until you don't make it anymore.
Chess is actually really easy to improve in - fast, trivially repeatable games, chess engines, lots of learning resources, objective rating system.
> I know that I do not have the brain to be a chess master
I'm not sure what exactly you mean by 'chess master', but for example getting into the top 10% of players is pretty easy. You are indeed incredibly unlikely to be one of the best, because for that you will have to dedicate your entire life to playing chess. But unless you literally want to be one of the best (top 1%+ of players), genetics will not limit you. You may take more time to get to any given level than someone gifted with more working memory or whatnot, but it's still doable.
> because I don't have enough working memory to keep the board in my head.
Because of humans' limited memory and processing power, we play games not by exhaustively analyzing but via building simplified models and heuristics so I don't see why you'd need to 'keep the board in your head'.
My belief comes my experience teaching math the past 20 years. I think we both agree that mentally disabled people can’t learn algebra. Do you have any evidence that these are the only people incapable of learning algebra? I have lots of anecdotal evidence that the set of people who can’t learn algebra is much larger than the set of mentally disabled people.
Do I require evidence to not believe something you've stated? Or do you require evidence to assert something you've stated?
I don't think the onus is on me to prove you wrong, so much as the onus is on you to prove that your experience is scientifically rigorous and representative of more than just your personal experience.
And to be clear, I don't doubt that many people "can't" learn calculus in the same way I "can't" run a marathon; we don't have the desire, discipline, and free time necessary to do the work required. This does not speak in any way towards our capacities to do so, only our desire.
So I suspect the vast majority of your students didn't have the sufficient desire to learn a lot more than they were incapable, and I think it's a critical distinction, because you can change people's motivations, but you cannot change their capacity.
There’s no onus on either of us to do anything. We are just strangers posting on the internet without pay or other compensation. Your experience and knowledge about teaching help me to decide the merits of your beliefs. It appears from my perspective that you know quite little about what you are posting. I know slightly more by having a lot of anecdotal evidence but this certainly doesn’t imply that I’m more likely to be right.
As stated in my original comment I’m quite liberal. In education we talk a lot about social conditions and their effects on education. It’s why teachers’ unions strongly support universal healthcare, school lunch programs, etc.
After 20 years on the job I’ve come to the conclusion that some can’t learn it. This isn’t controversial in some sense since we know retarded people can’t learn algebra. So what level of capability is necessary to learn it? I don’t know but I suspect it’s well above being mentally retarded.
I could be completely wrong but given your lack of experience in the topic my experience ought to at least make you pause a bit. In another comment you wrote about me:
… you're more likely ignoring the real-life circumstances your students find themselves in, and your colleagues are not.
Talk about making an assertion with no evidence! At least I waited 20 years before stating my absurd assertion.
It's clear to me that this belief has wormed its way into your identity somehow, and you're unable to discuss it in a detached and curious way. I'm sad for you that such is the case, but I am now more empathetic towards your colleagues who have to deal with your false assertion that some people are too stupid to learn the math you teach.
Also not for nothing, but "retarded" hasn't been a medical term for some time now, with institutions such as the AMA and SSA both replacing the term with "intellectual disability". The only remaining reasons you'd use it are either because you are thus uninformed, or you're signaling something...
It’s easier to type retarded than intellectually disabled on an iPad. I wrote the modern version in my original comments. Clearly I have been discussing this topic in a detached way. You’ve formed an image in your mind about me that isn’t supported by the evidence.
I’ll read whatever response you have but won’t comment further. I wish you well. Keep up the good fight!
You can use "cognitively impaired" as a more neutral and factual descriptor than either (and "cognitive impairment" as the generic term). It's also more broadly applicable. (I.e. If you got whacked in the head once too many while playing rugby, that might not count as "intellectual disability" according to some since it's not developmental. But it might interfere with learning math, so it's pretty indistinguishable in a practical sense!)
Thanks, I appreciate you taking the time to talk about a view you've held that's been unpopular with your colleagues.
I think you, and everyone who thinks like you, need to take a very hard look at what you've done to eliminate the environmental factors related to your viewpoint that some people aren't smart enough to learn something, because its extremely easy to trick yourself into thinking someone is incapable when in reality other factors abound.
As the article explains, if society refused to educate/feed/raise/nourish red-headed children, there would be a genetic correlation between red-headedness and intelligence.
How absolutely confident are you that you've accounted for every explanation besides the genetic one in determining why some people can't learn algebra?
Its kinda crazy how casual people are with their constraints on explantion when the topic is intellectual disparity. Person you're replying to claimed they weren't arguing for a gentic component to another reply but also doesn't do anything to filter any other factor than a congential threshold for learning math. This is absolutely why the Left is generally leary of this type of discussion because it is so easy to use the information for all sorts of ill-informed or cross purposes.
It's pretty clear you've never instructed a remedial math class.
My university would conditionally admit students who had a math score below a certain threshold on their ACT, I think it was 19 or something. Anyway, as part of their admittance criteria, they had to attend an after class lab for an additional hour an a half for a total of three hours per week dedicated to learning pre-algrebra. There were four modules where they would do some reading, work through some example problems through interactive software, and then have some homework to work through that was almost identical to the examples. Students would try and try and try and try to learn the material, and they would take days to work through the problems on their own (often with my guidance, giving pointers on how to think about the problems) to finish the module so they could take the quiz and pass the module. Near the end of the semester many students made appreciable progress, but for others the inability to retain and apply what they've spent so much time on results in tears, especially because they don't know if this requirement will keep them from being able to graduate.
Given infinite time, could these guys all have figured out pre-algrebra enough to pass? Maybe. But the amount of time it takes them to learn math concepts that are very easy for us means that it's entirely impractical to expect them to ever achieve proficiency in advanced mathematics.
> There were four modules where they would do some reading, work through some example problems through interactive software, and then have some homework to work through that was almost identical to the examples.
This just does not look like a universally effective way of teaching to me, irrespective of the topic. It's hardly any wonder that some people fell through the cracks if they were unfamiliar with the subject in the first place. What about leveraging stuff that's actually been tried and tested, like the Khan Academy videos and their automated interactive, school-like environment?
> And to be clear, I don't doubt that many people "can't" learn calculus in the same way I "can't" run a marathon; we don't have the desire, discipline, and free time necessary to do the work required. This does not speak in any way towards our capacities to do so, only our desire.
Your VO2max, max heart rate, and other factors appear to be significantly determined by genetics, and will absolutely contribute to your capacity to run a marathon. If it takes you a year of hard training, but it takes me a few practice runs a few weeks before, it's not fair to say we required the same amount of desire, discipline, or free time to succeed. I would instead say we had a very different capacity to run a marathon.
Are you saying that similar things could not possibly be true for learning math? Or really anything else that humans do?
I don't think any of those things contribute to whether or not I could complete a marathon, though they do contribute to how pleasant the experience would be. The same holds for learning calculus, or doing anything intellectually taxing. It's differing in difficulty for people, however it's not unachievable for anyone who isn't experiencing some kind of mental (or physical, in this marathon example) impairment.
All healthy people can learn calculus and run marathons, with varying degrees of success and effort, due to genetics and environmental factors.
I don't see how if you can agree that it requires varying degrees of effort for different people to run a marathon or learn calculus, that you then think it is impossible that some people won't be able to do those things. Just like the amount of effort required will be small for some, it will be impractically large for others. Even in the theoretical sense, there is only so much time in a day.
I also don't think "healthy" and (presumably) "not healthy" are useful categorizations. There are many people in that fuzzy in between area between "healthy" and "not healthy", for both physical and mental health.
Is it really worth so vigorously arguing the semantics of "some people are incapable of learning algebra" and "it requires an impractically large amount of effort for some people to learn algebra"?"
Yes, I think it's absolutely worth arguing the difference between "it's hard" and "it's impossible", because those are two fundamentally different things.
You can overcome difficulty, you cannot overcome (by definition) impossibility.
What you keep ignoring in what I'm saying however, is that I do think certain things are impossible for some people. I will never play in the NBA, for example, but that's a far cry from being "very good" at basketball.
Learning calculus and completing a marathon are not the point at which "healthy" (and yes, it's fuzzy, but precision is impossible on this topic) people are sometimes unable to do things. Winning a marathon and getting a Ph.D. in mathematics, I would acquiesce to your argument.
In other words, I think anyone can dabble in anything, but you do need an alignment of genetic and environmental circumstances to be in the top 1% of something. I could be argued into top 10%, but below that, it appears the data supports almost anyone being able to do almost anything, or at least well beyond whatever artificial lines we might draw to discourage people from achieving.
> I don't think the complexity of either rises to the level of "difficulty" that only gifted minds can comprehend
This is a strawman. They never said 'gifted', they just said that some people have it and some don't.
> deficiencies in one area can be made up for in other areas
Not what we're talking about. GP is probably great at things that aren't chess. That doesn't mean they're good at chess.
> Where genetic gifts are lacking, determination and perseverance can almost always make up the difference
Determination and perseverance aren't (at least partially) genetic gifts?
Honestly I think 'genetics' are a bit of a red herring here anyway because the real underlying assertion being challenged is that all humans have equal potential in all things (which anyone who has ever taught any mildly difficult topic will know is trivially disprovable). The precise mechanism whereby innate human potential differs isn't important if you can't even agree that it differs.
Your belief should probably be considered heretical, even among your conservative colleagues, because your colleagues should recognize how deficiencies in one area can be made up for in other areas, sometimes (but not always) at a higher efficiency cost.
Where genetic gifts are lacking, determination and perseverance can almost always make up the difference, especially at a basic algebra/calculus level.