I don’t think there is a clear problem statement anywhere about what is wrong with math (or any other) education, or what “good enough” looks like. Almost nobody writing about this really accepts the differing abilities, motivations, and social support (parents and friends) of the students. There CAN BE no single solution—the variance is just too great.
Personally, I focus on getting the most out of the top potential learners, which DOES lead to research-ey theory of education, as it is there to identify and encourage future researchers and advanced thinkers. This portion of education leaves MANY students behind, because that’s not what they need, and not how they interact with educators. I think there needs to be another tranche of “smart, well-supported, non-academically focused” students, getting enough math to inform their daily life and help develop a numerate mindset. I don’t know how to move the less-well-supported and less-well-equipped (regardless of reason) into the well-supported group.
This is one of the problems that can’t be discussed very well in public, as it’s currently outside of mainstream Overton window to admit that there is a pretty large cognitive and motivational variance among humans. In fact, I feel it’s important to state that I don’t think this variance is necessarily ingrained or permanent across generations or across an individual’s lifetime. But I do think it’s real.
I have a strong intuition that we are far from the Pareto frontier. For most kids, math is suffering, and at the end they do not remember almost anything. Maybe we could teach them more, maybe we couldn’t… but if it’s the latter, then at least we could make them suffer less.
There is probably a lot of signaling involved. By making kids needlessly suffer, we signal that we care about their future well-being. By suffering, kids signal their diligence. Maybe the math is made difficult on purpose, because difficult math more clearly separates the more talented kids from the less talented ones.
I agree that people should pay way more attention to the differences in IQ. I also think that people often use it as a lazy excuse. (“Hey, most kids don’t understand your lecture.” “Not my problem, large cognitive variance.”—conveniently ignores the fact that the kids who didn’t have a problem with the lecture were the ones who already knew all of that from some other source.)
Remember the “camel has two humps” paper about programming aptitude? At first, it seemed that only human nature denialists could disagree with the harsh truth. And then it didn’t replicate. What I am trying to say is that although ignoring the cognitive variance is popular, it is also popular to overcompensate in the opposite direction and blame basically everything on genetics, even when something is clearly wrong.
(For example, some people have told me that teachers didn’t explain to them that “fractions” and “division” are the same thing, only written differently. You can’t blame that on a difference in IQ, because those people then successfully learned skills such as adding fractions, they just never clearly understood what they were doing; they just operated the numbers mechanically according to the rules they learned.)
(This summer I had a lecture for a group of people about “how I would explain complex numbers to an 8 years old child”. After the lecture, several math teachers thanked me, saying that this was the first time they actually understood the concept. I am currently rewriting it as a blog post… that, and some research related to that, was actually what triggered me to write this post.)
Or, you know, people say how much they learned from some educational YouTube videos. Okay then, why don’t we make some of those videos a part of school? I mean, completely literally, why couldn’t we have each week two hours of “video watching lessons”? Perhaps make it elective, like you can either watch a math video from 3Blue1Brown or a history video by Sabaton (and then of course you can watch the other one at home, if you want to). Just one example of what seems to be a low-hanging fruit. Or when kids learn numbers from 1 to 10, why don’t we let them spend a lesson or two solving easy Sudoku?
What I am trying to say is that although ignoring the cognitive variance is popular, it is also popular to overcompensate in the opposite direction and blame basically everything on genetics, even when something is clearly wrong.
Oh, quite! I tried to make that clear in my comment, but maybe it’s not possible to point it out without the assumption that one is on the complete opposite end. Sad, really—the multiple dimensions of ability, willingness, context, and possible improvements are too complicated to use a one-factor model.
“conveniently ignores the fact that the kids who didn’t have a problem with the lecture were the ones who already knew all of that from some other source.”
This is definitely not true in general and probably a rare case. N=1 of course, but I never had problems with maths lectures (or any other lectures) and I never was in the situation of knowing all of the maths before the lecture (I usually knew history and physics lessons in advance though). And it’s the same thing with my current students : even the best ones are clearly unfamiliar with the material I cover.
I think the lesswrong crowd has in general a very unusual experience with both school and maths, even compared to the average gifted maths student. Beware of the typical mind fallacy.
I don’t think there is a clear problem statement anywhere about what is wrong with math (or any other) education, or what “good enough” looks like. Almost nobody writing about this really accepts the differing abilities, motivations, and social support (parents and friends) of the students. There CAN BE no single solution—the variance is just too great.
Personally, I focus on getting the most out of the top potential learners, which DOES lead to research-ey theory of education, as it is there to identify and encourage future researchers and advanced thinkers. This portion of education leaves MANY students behind, because that’s not what they need, and not how they interact with educators. I think there needs to be another tranche of “smart, well-supported, non-academically focused” students, getting enough math to inform their daily life and help develop a numerate mindset. I don’t know how to move the less-well-supported and less-well-equipped (regardless of reason) into the well-supported group.
This is one of the problems that can’t be discussed very well in public, as it’s currently outside of mainstream Overton window to admit that there is a pretty large cognitive and motivational variance among humans. In fact, I feel it’s important to state that I don’t think this variance is necessarily ingrained or permanent across generations or across an individual’s lifetime. But I do think it’s real.
I have a strong intuition that we are far from the Pareto frontier. For most kids, math is suffering, and at the end they do not remember almost anything. Maybe we could teach them more, maybe we couldn’t… but if it’s the latter, then at least we could make them suffer less.
There is probably a lot of signaling involved. By making kids needlessly suffer, we signal that we care about their future well-being. By suffering, kids signal their diligence. Maybe the math is made difficult on purpose, because difficult math more clearly separates the more talented kids from the less talented ones.
I agree that people should pay way more attention to the differences in IQ. I also think that people often use it as a lazy excuse. (“Hey, most kids don’t understand your lecture.” “Not my problem, large cognitive variance.”—conveniently ignores the fact that the kids who didn’t have a problem with the lecture were the ones who already knew all of that from some other source.)
Remember the “camel has two humps” paper about programming aptitude? At first, it seemed that only human nature denialists could disagree with the harsh truth. And then it didn’t replicate. What I am trying to say is that although ignoring the cognitive variance is popular, it is also popular to overcompensate in the opposite direction and blame basically everything on genetics, even when something is clearly wrong.
(For example, some people have told me that teachers didn’t explain to them that “fractions” and “division” are the same thing, only written differently. You can’t blame that on a difference in IQ, because those people then successfully learned skills such as adding fractions, they just never clearly understood what they were doing; they just operated the numbers mechanically according to the rules they learned.)
(This summer I had a lecture for a group of people about “how I would explain complex numbers to an 8 years old child”. After the lecture, several math teachers thanked me, saying that this was the first time they actually understood the concept. I am currently rewriting it as a blog post… that, and some research related to that, was actually what triggered me to write this post.)
Or, you know, people say how much they learned from some educational YouTube videos. Okay then, why don’t we make some of those videos a part of school? I mean, completely literally, why couldn’t we have each week two hours of “video watching lessons”? Perhaps make it elective, like you can either watch a math video from 3Blue1Brown or a history video by Sabaton (and then of course you can watch the other one at home, if you want to). Just one example of what seems to be a low-hanging fruit. Or when kids learn numbers from 1 to 10, why don’t we let them spend a lesson or two solving easy Sudoku?
Oh, quite! I tried to make that clear in my comment, but maybe it’s not possible to point it out without the assumption that one is on the complete opposite end. Sad, really—the multiple dimensions of ability, willingness, context, and possible improvements are too complicated to use a one-factor model.
“conveniently ignores the fact that the kids who didn’t have a problem with the lecture were the ones who already knew all of that from some other source.”
This is definitely not true in general and probably a rare case. N=1 of course, but I never had problems with maths lectures (or any other lectures) and I never was in the situation of knowing all of the maths before the lecture (I usually knew history and physics lessons in advance though). And it’s the same thing with my current students : even the best ones are clearly unfamiliar with the material I cover.
I think the lesswrong crowd has in general a very unusual experience with both school and maths, even compared to the average gifted maths student. Beware of the typical mind fallacy.