I think the more people can learn algebra than OP implies, but as a former math educator my experience also indicates some people are genetically limited in mathematics. For reference, my specialty was working with very remedial students and only ever had a few students not make progress (~2%) but I do think that small percent was genuinely hopeless and I don't say that lightly. I have a pet theory as to why those students could not do algebra that that you might find convincing:
To start, I believe that the idea of "working memory" is largely valid. Think of it as the number of distinct ideas you can hold in your head at once, sort of like trying to hold a phone number in your head when you've just heard it for the first time.
The general consensus in psych is that this number for the average person is in the single digits and is relatively static in adulthood till a decline in old age. It's been my observation that people with really incredibly small working memory cannot do algebra. The amount of numbers/ideas held in their head is too large, and multiple students in this group described the experience of attempting an algebra problem as feeling like sand constantly slipping through their fingers.
Many of these students grew up in rich neighborhoods with good parents. They had most advantages you can imagine, seem reasonably intelligent when you talk to them, but Algebra will always be beyond them.
My dad was a math and physics teacher and he had a very similar theory around the working memory. However he also thought that the problem for those students that had difficulties with math was that the working memory was often filled up with other stuff than the math that they were working on. It could be anything from difficulties to concentrate to problems with holding the pen correctly and therefore the working memory was overflowed.
This describes my experience with ADHD except your head is filled with impulsive thoughts (noise), so you struggle to string together a bunch of numbers in a logical forward progression through a formula, which also induces cognitive load when recalling the formula to your working memory when it is overwritten by the impulsive thoughts.
I can get around this by brute forcing the numbers into the formulas on paper and going through each step slowly but surely, even steps that people can do mentally. I just need significantly more time than my peers to finish.
Taking stimulants and increasing my dopamine levels across my synapses alleviates much of the working memory deficits I outlined.
I wonder how many of these students that struggle were just undiagnosed ADHD and not deficient in an low IQ mental handicap sense.
> I wonder how many of these students that struggle were just undiagnosed ADHD and not deficient in an low IQ mental handicap sense.
I've got ADHD that was undiagnosed until a decade after finishing my degree, and maths was my favourite - and best - subject all the way from age 4 through to doing solutions to Einstein's equations involving 10+ A4 pages (both sides) of tensor mechanics for my final year project. My results across that time showed the more maths in a subject the better I did, the more writing it had the worse I did.
Doing maths is like reading is for me - an external cognitive structure that I can follow to make my own brain calm down while in that process. While I don't do much maths nowadays, I literally read whenever I'm not actively doing something else - I read a page or two of my book in-between clicking reply to your comment and starting typing this reply.
Maths involves a lot of "muscle memory" once you get past the initial hump. But it's often poorly taught at an early age to the level of inducing near-phobic levels of discomfort with it which to me seems strange. But I don't think there's any extra issue with having ADHD and maths
ADHD is very comorbid with dyslexia, dyspraxia, dysgraphia and dyscalculia - around 20–60% of people with ADHD also have one or more learning difficulties. Dyscalculia affects as many people as dyslexia, but dyslexia is far more well-known and more likely to get diagnosed and helped with. There's a nice summary box of typical symptoms of dyscalculia in this paper:
> To start, I believe that the idea of "working memory" is largely valid. Think of it as the number of distinct ideas you can hold in your head at once, sort of like trying to hold a phone number in your head when you've just heard it for the first time.
Interestingly, testing myself with the Wechsler reverse-digit-span test, I found very large improvements in working memory from taking 20-minute afternoon naps and from modafinil. I'm not claiming everyone is alike, of course, but working memory is definitely not as fixed as height and eye color.
Specifically with regard to mental math, I find I can do a lot more when I'm lying in bed in a dark room than when I'm in an uncomfortable chair in a noisy cafe with people talking to me. Or, for that matter, classroom.
> Many of these students grew up in rich neighborhoods with good parents. They had most advantages you can imagine, seem reasonably intelligent when you talk to them, but Algebra will always be beyond them.
I tried marijuana once and found the opposite effect: when I was high, by the time I got to the end of saying a sentence, I couldn't remember how it had begun. But people report that I seemed like I was conversing normally; if they didn't know me, they wouldn't have realized anything was off. I wonder if these folks were experiencing something similar all their lives?
> I found very large improvements in working memory from taking 20-minute afternoon naps
Sure. My ability to do anything also drops to near zero if I don't sleep enough. Yet I have not found unlimited increases in capability if sleep more and more...
> modafinil
Yeah, drugs are a quite different beast. Doping happens when individual athlete realizes they have hit the limits what their "natural" biology can do and yet still are not going to win the competition.
Yeah, the drawback of modafinil is that it seems to reduce my ability to recall things from long-term memory. Also, it makes it hard to switch tasks when it's necessary.
I don't think I'm suffering from sleep deprivation. Certainly it is true that if you take enough stimulants your performance will worsen! In fact, if you take enough stimulants you'll die and your performance will be zilch!
> To start, I believe that the idea of "working memory" is largely valid.
Wait, are there mainstream schools of thought where working memory isn't considered valid? I'm a layman but I suffer from ADHD, and very much notice that my working memory fluctuates with my attention span (from lack of sleep/stress/etc).
Google's of no help to me, but I remember a story of some educators, looking at some kids who didnt go to school but worked selling concessions, but in turn, were actually quite good at math. They couldn't answer math questions when written out on a worksheet, but they could do the exact same questions when presented in the form of a complex order. (Double digit multiplication and summation isn't the same as algebra, but being able to do that implies a large working memory, which is claimed is the barrier to learning algebra.)
I do have poor working memory and I did struggle with algebra, especially with copying wrong sign from row to row and things like that.
But I did make paper sort of my working memory and when doing algebra I felt that I was just the very resource limited CPU that executed the instructions from paper-memory.
Algebra always made me feel like I was doing some mindfull exercise where I had to empty my mind, follow the paper script and hope I didn't mess anything while switching from row to row of calculations.
Even today, as a programmer, I struggle to remember class or function names, I just empty my mind and am really good at searching stuff in code.
I read this paragraph in a paper on ADHD and learning disabilities literally after replying to a comment above yours...
> There is also evidence of domain-specific cognitive deficits that contribute to specific learning-related disabilities. For example, phonological processing difficulties have been found in children with poor reading performance, whether or not they also exhibited problems with ADHD symptoms or math, but not in children with deficits in ADHD symptoms or math only. Similarly, both with and without a reading deficit, children with ADHD symptoms exhibit significantly impaired object naming and behavioral inhibition, and math-disabled groups demonstrate visuospatial and numerical processing deficits, while those with only reading problems sometimes do not.
There's significant co-morbidity between learning disorders and ADHD, and despite dyscalculia being as common as dyslexia (~3-7% of the population) it's a lot less well-known, isn't as frequently diagnosed and there are fewer tools to help people with it. It would be very possible to have both ADHA and dyscalculia, given you used symptoms listed for both that are almost word-for-word identical with those in the paper...
I had no problem with reading or abstract thinking or grasping mathematical concepts.
My problem was that oftentimes I would screw up calculations for a math problem and only get 7/10 score on it (the reasoning was mostly right, but the answer was not).
Well, I’m certain cockroaches can’t learn calculus. Neither can apes or monkeys. Some humans can. Therefore I think that brain composition has something to do with learning calculus. There is variation in said brain composition amongst humans. It seems reasonable to think that the set of all people is not a subset of all things that can learn calculus.
I haven’t performed any genetics tests but brain composition is partly determined by genetics so genetics might play role, right?
I always wondered if the teachers understood how much effort people put in. Especially in something like math, where you can't just waffle.
I got the feeling as a kid that a lot of my classmates would just give up. I mean they're sitting there acting like they're concentrating on math, but they aren't. In other classes intermittent attention is enough, you pick up social cues and repeated stories that you've heard, and poof you have a history essay. With math there's a need to have all the steps.
How do you control for that as a teacher observing the kids? How do you know whether a kid has actually tried to learn stuff at home?
I still remember my brother was baffled at how I got top grades in math when he never saw me studying. Of course I tended to do that when he wasn't around.
> How do you control for that as a teacher observing the kids?
The only solution I ever found was not having more than 3 students at a time, not an option for most teachers; I spent most of my career working with 1-3 students as a result. With that few students you can carefully observe the mistakes students are making and ask individual questions about their mental state. Experience will eventually tell you to differentiate students not putting in effort from students who are so lost that they're just flailing and hoping something sticks.
With more students, my experience was that both my attention became too split to give students the kind of careful diagnosis to control for effort.
I didn’t imply anything about the number of people who can’t learn algebra other than that I think the number of such people exceeds the number of intellectually disabled people. I used the word intelligence and put this in quotes because intelligence is not really definable but the word does convey a sense of what I mean. One must have a certain level of cognitive ability to learn something. Where that level is for a given topic I can’t say.
You've offered some evidence as to why you believe some individuals are incapable of learning algebra, but you haven't offered an explanation as to why what you observed is necessarily the result of genetics, there could be dozens of additional factors at play here which are not being considered.
HN crowd doesn't think twice about people who try to run DOOM on a stack of pennies, but are ready to give up before trying to get algebra to run on someone with less working memory. It's weird how different the attitude is.
Have you ever been a teacher? Even a teaching assistant?
I have. Some people are dumb.
You try teaching these people (anecdotally they're less than 1% of the population, with no obvious markers, so you have to actually find one first) and then come back and tell me how it's like "getting algebra to run on someone with less working memory". If I wanted to continue that analogy, I might say that the task is like getting algebra to run on someone whose brain has a power supply that randomly shorts out and spends half its time browned out.
Because some people out there are just not easy to teach.
I'm not sure I understand your comment, this is exactly what I spent hundreds of hours of my life trying to do. Was I unclear that my entire educational career was being the person who got through to students that traditional education was giving up on?
I disagree - we have excellent tooling avaliable to cram and compile code into smallest CPUs, but our teaching methods are still the same as they were in 1800's
I think this reveals a lot of ignorance on the huge amounts of different teaching methodologies and techniques that have risen and fallen over the last 100 years.
To start, I believe that the idea of "working memory" is largely valid. Think of it as the number of distinct ideas you can hold in your head at once, sort of like trying to hold a phone number in your head when you've just heard it for the first time.
The general consensus in psych is that this number for the average person is in the single digits and is relatively static in adulthood till a decline in old age. It's been my observation that people with really incredibly small working memory cannot do algebra. The amount of numbers/ideas held in their head is too large, and multiple students in this group described the experience of attempting an algebra problem as feeling like sand constantly slipping through their fingers.
Many of these students grew up in rich neighborhoods with good parents. They had most advantages you can imagine, seem reasonably intelligent when you talk to them, but Algebra will always be beyond them.