I mostly agree with your points, but in practice my experience teaching mathematics is that often by *trying* to teach more kids math we make sure only the gifted really understand it. The problem is the incentives favor moving more to rote learning when you try to get more students to succeed and the end result is often only the very gifted actually can extract real conceptual understanding from the class.
The problem is that even if you're intelligent and conscientiousness if you hate math and thinking about it makes you feel frustrated and miserable it's exceptionally difficult to engage in the kind of hypothesis generation and testing (ohh maybe it works like this) needed to get a real conceptual understanding. If the teacher asks questions that can't be done by rote and test real conceptual understanding those students just can't do them.
OTOH those students can memorize rote rules with enough effort and pain. The net result is that the incentives for the school, teachers and most students are to just learn those rote rules. The very gifted students can still extract understanding but you've lost all the kids in the middle who could have excelled in a class that asked conceptual questions on exams and required real understanding. But I fear that in order to reach those students your incentives have to allow for the fact that you can't give a backup rote mechanism for doing well (so some motivated students ..even with pushy parents.. won't succeed).
So I sorta agree with you in principle but I fear that when you give this as a policy goal the way it trickles down into the classroom is to incentivize teachers to increase the fraction of students who succeed in meeting some testing threshold and that often means presenting the class with a list of rote tricks. So instead of getting say (in say a high achieving suburban class) 70% of kids who leave with real understanding you get 20% with real understanding and 90% who can manage to the computations (and forget them once the class ends).
I think this perspective, combined with the research around deliberate practice, does result in policy recommendations that will have a lot of enemies. Far more hours of the school day need to be used for guided practicing of math.
The way more people learn math is to have more time doing exercises in a deliberate practice framework (eg JUMP math is one example, khan academy another). But homework is a bad tool for this because real time correction and redirection is essential. The goal is to be practicing at the edge of ability, which means it’s easy to get blocked without a guide.
This does mean less time for other topics. From my perspective, there does not need to be as much time spent on study that is mostly memorizing names and facts.
But I expect this recommendation to provoke a lot of disagreement.
I’m going to disagree with the parents drilling is what differentiates kids.
I’ve raised 9-kids. Yep. 9.
The simple fact is that some of them pick up on math and learn it easier than other kids. Completely independent of what parents did or didn’t do.
What the real issue is, is math/science curriculum is often poorly taught. The natural kids will learn it anyway. It’s the other kids who suffer.
To summarize better math instruction will not eliminate the gaps. It might not even close the gaps. But it certainly raise the average and baseline for everyone.
thesis here is a bit self-contradictory no? on one hand it's wrongheaded to focus on natural talent, education is not about screening etc. but on other it mentions how school choice is not very effective at improving grades, the practice of shunting kids into accelerated math classes based on test scores hasn’t been shown to have a long-lasting effect etc. which basically means education doesn't actually improve your abilities, i.e. it's after all about screening and natural talent, you just may need to screen both on smarts and motivation rather than former element only.
Talking about motivation, these kids in Asia aren't by and large intrinsically more self-motivated. Their motivation comes from outside-in, in form of parental pressure, social pressure (everyone else is forced to go to cram schools after regular school so you go too instead of playing videogames) and economic pressure (yes engineers in Taiwan at TSMC don't make that much by US tech industry standards, but given that many starting salaries for college grads in TW are sub US federal min wage (you read that right - here is a random link i just googled that gives some more details - https://forums.hardwarezone.com.sg/threads/how-come-taiwan-uni-grad-starting-salary-so-low-nt-30000-sgd-1500.6629812/ and there is constant hand-wringing about the topic on TV but still little seems to be changing) these TSMC engineers are still getting paid a lot relatively speaking).
Math is just inherently a hard subject and boring to probably 90%+ of the population no matter how engaging and fun you try to make it out to be (I was gifted at math when young and am home-schooling my kids now, pretty sure I'm way above regular school teacher in terms of motivation, competence and attempts at fun/engagement, it's still a slog) so the only way to achieve broad math competence in the population beyond the sub-segment of vaguely on the spectrum kids that are naturally attracted to it is through coercion. I just can't see any appetite to enforcing that broadly at any societal level in America (even assuming it would be a desirable thing, which am ambivalent about. e.g. still debating myself about my own kids how hard I want to push them in that direction, particularly with the one that isn't showing much natural talent towards STEM subjects).
Hear, hear! AS a graduate of a gifted math program (I was never Putnam-exam-level good) I am appalled that very good math education for the few is bought at the cost of appalling math education for everybody else. People should understand that math is an actual creative activity.
And something I do not do enough of: People should be confident in being able to run the numbers themselves if they have to think out their positions on policy issues.
I think the big thing is we just have to make school harder. Professors need to push their kids more.
I always found the people talking about pushing Algebra to 9th grade really weird. When I was in school we started learning Algebra in 6th grade - with the proper preparation, you could probably start kids with it in 5th. If you have a good grasp of your multiplication and division tables, you can get started on algebra. We should be seeing how early we can do Algebra and math rather than fumbling about how late we should push it.
Third, California graduates more than enough Master's and PhD electrical engineering, physics and material science students to work in the one large semiconductor fab (Intel) that exists in California. California used to have more fabs, but most fabs in the US today are in New York State, Vermont, Oregon, New Mexico and Arizona. The most leading edge fabs in the world are in Taiwan. Most of the integrated circuits in mobile phones are made in fabs in Taiwan, not California.
I'm not saying that California shouldn't be doing more to focus on the core curriculum in the K-12 system. That being said, the small number of fabs remaining in California have PhDs beating down the door to work in them.
I think you’re mistaken. They need employees with high IQs to solve complex problems. You could perhaps take someone with an IQ of 100 and teach them enough math and physics to pass some sort of fab entrance exam. Maybe. They would still be too stupid to do the job.
Noah, I’m curious, in your experience have you run into folks who are trying their best and have the credentials but are just too stupid to do the job? It’s fairly common in my line of work. People have the proper educational background, they interview well and they are hired. But as training starts it soon becomes obvious that they just aren’t capable of picking things up as fast as we need them to.
A lot of folks seem unable to accept the idea of too stupid. If you have good enough teachers and the student studies hard enough anyone can do learn anything. Sadly that’s not how it works.
An important paper here https://www.pnas.org/doi/10.1073/pnas.2221311120 shows that differences in student performance are almost entirely related to prior exposure rather than innate ability. "These results provide a challenge for learning theory to explain this striking similarity in student learning rate. They also suggest that educational achievement gaps come from differences in learning opportunities and that better access to such opportunities can help close those gaps. " Using a data set of 200 observations on each of almost 7,000 students they find that initial performance varies substantially but earning rates are virtually constant regardless of the initial level of performance.
Very Interesting article... let me make three somewhat disconnected points:
1) STEM Education: The current education model is not connected to the times. The world needs an on-demand highly scalable delivery structure for education (including STEM). The cost of this should be near free. In this model, education resources are pulled based on need. There is nothing like trying to solve real problems to motivate learning something. Note.. this model exists for adult education today.
2) Grades vs Learning: Rote disconnected learning as we find in a lot of forced STEM programs is not helpful. Richard Feynman made this observation with the contrast between the Brazilin and US education system. I myself have seen it with Chinese and Indian education systems in STEM.
3) Mass vs 3-sigma: Is it better to be better at Mass STEM education or 3-sigma ? In general, technology invention is more like basketball. 3-sigma individuals outrank a mass of engineers. Finding, helping, and hoarding the worlds' best talent is very important. Also, mass "jobs" get designed out by technology over time by the 3-sigma solutions.
Finally, the most important point... freedom is really important. I don't mean freedom in some egalitarian nice-to-have sense. I mean freedom-of-thought in the problems to attack and the freedom-of-thought in the economy to consume goods/services (the plasticity of an economy).
The subtle consequence of all the judgement connected with grades[useless in my view] as well as the constraining thought process of the latest technological thrust (STEM as an example) has an impact on innovation (local and global). At the end, the rate of innovation is the only thing that really matters.
To enable the above, you need three things:
1) a service oriented education sector... as I said on-demand and early free.... [believe it or not..we are almost there...and I don't mean the establishment in education]
2) encouragement of citizens to solve problems around their passions [bottom-up innovation]
3) a rich "soil" of financial support for experimentation of ideas
Note.. for all of this to work, there is a severe need for basic values: trust, earnestness, hard-work..etc. This perhaps maybe the most important goal of the k-4 system... and of course parents. Part of the way the US works is because it has large elements of the above three.
Overall, it would be good not to destroy what works well in the zeal to follow some mass model.
Ii know you used the example of fab factories because they are shiny and high tech, but it is really a lot wider than that. I work for a chemical company. Our process is highly automated. The control computers can run the process better than a person could, because the computer doesn’t get bored. But while the computer is doing it’s thing, heat and dirt and acid are working to tear the equipment apart. We need skilled operators who can monitor the control graphs and go though the plant to monitor all of the equipment to handle the inevitable problems that the computers can’t handle. This takes basic math skills and plenty of critical thinking skills to do successfully. The manufacturing jobs that come back to the US in the future will be very different from the ones that moved away in the 80s and 90s.
One question I’d love an answer from opinion writers of pieces like this is:
“What level of math does the author believe the average American is capable of attaining?”
Because without an answer to that fundamental question, I’m always left wondering “why bother?” after reading pieces like this.
I’m currently convinced that the average American, heck, the average human, is bad at math (I most definitely include myself in this category!) and will not be capable of learning math to the level required for all of these jobs of the future.
Given the strong bifurcation of math ability required for jobs like those highlighted here, and most other occupations that require little more than middle school math, where is the value in raising the level of math attainment for the fat slice of the middle in the country?
A nation where everyone learns basic calculus is likely not going to be materially better off than one where everyone can do basic arithmetic so long as they both have enough people to fill their extremely advanced positions. That’s where the entirety of our focus should be, not on raising our median math education levels to a still unsatisfactory level.
I’ve never heard anyone argue for less STEM to increase egalitarianism -- citations would be great. We do need STEM but we need humanities just as much. Complete and well-rounded education is vital. Start by federally funding it so our southern citizens can get a decent education.
My youngest daughter was diagnosed as an Aspie in 2d grade. She was smart and literally hated middle school and its social games. 6 weeks into 6th grade she was still repeating math she knew (I had supplemented her math education to make sure she knew it), so I had her jumped to 7th grade math. In 8th grade I had her do 50% online education, took her to the high school (along with numerous other parents) for her math as the middle school didn't offer that level of algebra, and had her do her science, art/music, and physical education at her middle school. I had her skip 8th grade and do geometry by correspondence over the summer after 7th grade. She did Algebra 2 in 9th grade, precalc by correspondence over the next summer, and then calc for college credit in 10th grade. She dropped out of high school then and did early admissions to the state university, where she did her BS and MS in civil and structural engineering. I had to give the principals hell - they kept trying to get in the way. I told the principals that I did not care about state education requirements or school requirements - I was driving her education against college admission requirements - and high school graduation was not always required. She HATED the social scene in school, particularily in middle school.
My son wasn't quite as dilligent or mathematically talented, but he too did calculus in what would have been 11th grade - he did Running Start. and almost graduating with his associates. He did a Business - MIS degree in 7 quarters and then a MS in MIS - Data Security.
Too many of the Running Start students are slackers looking to escape the schedule of high school and then taking the easiest classes to graduate rather than stacking up transfer credits (and putting in the time to master the material) to prepare for the rest of their undergraduate education.
I am old and the kids knew that they didn't have backup - they had to be self supporting as soon as they graduated. We did not know if I would be forced to retire before they graduated - ageism in the tech industry is the rule, not the exception.
I mostly agree with your points, but in practice my experience teaching mathematics is that often by *trying* to teach more kids math we make sure only the gifted really understand it. The problem is the incentives favor moving more to rote learning when you try to get more students to succeed and the end result is often only the very gifted actually can extract real conceptual understanding from the class.
The problem is that even if you're intelligent and conscientiousness if you hate math and thinking about it makes you feel frustrated and miserable it's exceptionally difficult to engage in the kind of hypothesis generation and testing (ohh maybe it works like this) needed to get a real conceptual understanding. If the teacher asks questions that can't be done by rote and test real conceptual understanding those students just can't do them.
OTOH those students can memorize rote rules with enough effort and pain. The net result is that the incentives for the school, teachers and most students are to just learn those rote rules. The very gifted students can still extract understanding but you've lost all the kids in the middle who could have excelled in a class that asked conceptual questions on exams and required real understanding. But I fear that in order to reach those students your incentives have to allow for the fact that you can't give a backup rote mechanism for doing well (so some motivated students ..even with pushy parents.. won't succeed).
So I sorta agree with you in principle but I fear that when you give this as a policy goal the way it trickles down into the classroom is to incentivize teachers to increase the fraction of students who succeed in meeting some testing threshold and that often means presenting the class with a list of rote tricks. So instead of getting say (in say a high achieving suburban class) 70% of kids who leave with real understanding you get 20% with real understanding and 90% who can manage to the computations (and forget them once the class ends).
I think this perspective, combined with the research around deliberate practice, does result in policy recommendations that will have a lot of enemies. Far more hours of the school day need to be used for guided practicing of math.
The way more people learn math is to have more time doing exercises in a deliberate practice framework (eg JUMP math is one example, khan academy another). But homework is a bad tool for this because real time correction and redirection is essential. The goal is to be practicing at the edge of ability, which means it’s easy to get blocked without a guide.
This does mean less time for other topics. From my perspective, there does not need to be as much time spent on study that is mostly memorizing names and facts.
But I expect this recommendation to provoke a lot of disagreement.
I’m going to disagree with the parents drilling is what differentiates kids.
I’ve raised 9-kids. Yep. 9.
The simple fact is that some of them pick up on math and learn it easier than other kids. Completely independent of what parents did or didn’t do.
What the real issue is, is math/science curriculum is often poorly taught. The natural kids will learn it anyway. It’s the other kids who suffer.
To summarize better math instruction will not eliminate the gaps. It might not even close the gaps. But it certainly raise the average and baseline for everyone.
thesis here is a bit self-contradictory no? on one hand it's wrongheaded to focus on natural talent, education is not about screening etc. but on other it mentions how school choice is not very effective at improving grades, the practice of shunting kids into accelerated math classes based on test scores hasn’t been shown to have a long-lasting effect etc. which basically means education doesn't actually improve your abilities, i.e. it's after all about screening and natural talent, you just may need to screen both on smarts and motivation rather than former element only.
Talking about motivation, these kids in Asia aren't by and large intrinsically more self-motivated. Their motivation comes from outside-in, in form of parental pressure, social pressure (everyone else is forced to go to cram schools after regular school so you go too instead of playing videogames) and economic pressure (yes engineers in Taiwan at TSMC don't make that much by US tech industry standards, but given that many starting salaries for college grads in TW are sub US federal min wage (you read that right - here is a random link i just googled that gives some more details - https://forums.hardwarezone.com.sg/threads/how-come-taiwan-uni-grad-starting-salary-so-low-nt-30000-sgd-1500.6629812/ and there is constant hand-wringing about the topic on TV but still little seems to be changing) these TSMC engineers are still getting paid a lot relatively speaking).
Math is just inherently a hard subject and boring to probably 90%+ of the population no matter how engaging and fun you try to make it out to be (I was gifted at math when young and am home-schooling my kids now, pretty sure I'm way above regular school teacher in terms of motivation, competence and attempts at fun/engagement, it's still a slog) so the only way to achieve broad math competence in the population beyond the sub-segment of vaguely on the spectrum kids that are naturally attracted to it is through coercion. I just can't see any appetite to enforcing that broadly at any societal level in America (even assuming it would be a desirable thing, which am ambivalent about. e.g. still debating myself about my own kids how hard I want to push them in that direction, particularly with the one that isn't showing much natural talent towards STEM subjects).
Hear, hear! AS a graduate of a gifted math program (I was never Putnam-exam-level good) I am appalled that very good math education for the few is bought at the cost of appalling math education for everybody else. People should understand that math is an actual creative activity.
And something I do not do enough of: People should be confident in being able to run the numbers themselves if they have to think out their positions on policy issues.
I think the big thing is we just have to make school harder. Professors need to push their kids more.
I always found the people talking about pushing Algebra to 9th grade really weird. When I was in school we started learning Algebra in 6th grade - with the proper preparation, you could probably start kids with it in 5th. If you have a good grasp of your multiplication and division tables, you can get started on algebra. We should be seeing how early we can do Algebra and math rather than fumbling about how late we should push it.
Noah, a few facts.
First, I note that you switched out of STEM and went into Economics. Why would that be?
Second, the STEM Crisis is a Myth.
https://spectrum.ieee.org/the-stem-crisis-is-a-myth
Third, California graduates more than enough Master's and PhD electrical engineering, physics and material science students to work in the one large semiconductor fab (Intel) that exists in California. California used to have more fabs, but most fabs in the US today are in New York State, Vermont, Oregon, New Mexico and Arizona. The most leading edge fabs in the world are in Taiwan. Most of the integrated circuits in mobile phones are made in fabs in Taiwan, not California.
I'm not saying that California shouldn't be doing more to focus on the core curriculum in the K-12 system. That being said, the small number of fabs remaining in California have PhDs beating down the door to work in them.
Wait, you have a physics degree?
I think you’re mistaken. They need employees with high IQs to solve complex problems. You could perhaps take someone with an IQ of 100 and teach them enough math and physics to pass some sort of fab entrance exam. Maybe. They would still be too stupid to do the job.
Noah, I’m curious, in your experience have you run into folks who are trying their best and have the credentials but are just too stupid to do the job? It’s fairly common in my line of work. People have the proper educational background, they interview well and they are hired. But as training starts it soon becomes obvious that they just aren’t capable of picking things up as fast as we need them to.
A lot of folks seem unable to accept the idea of too stupid. If you have good enough teachers and the student studies hard enough anyone can do learn anything. Sadly that’s not how it works.
An important paper here https://www.pnas.org/doi/10.1073/pnas.2221311120 shows that differences in student performance are almost entirely related to prior exposure rather than innate ability. "These results provide a challenge for learning theory to explain this striking similarity in student learning rate. They also suggest that educational achievement gaps come from differences in learning opportunities and that better access to such opportunities can help close those gaps. " Using a data set of 200 observations on each of almost 7,000 students they find that initial performance varies substantially but earning rates are virtually constant regardless of the initial level of performance.
Very Interesting article... let me make three somewhat disconnected points:
1) STEM Education: The current education model is not connected to the times. The world needs an on-demand highly scalable delivery structure for education (including STEM). The cost of this should be near free. In this model, education resources are pulled based on need. There is nothing like trying to solve real problems to motivate learning something. Note.. this model exists for adult education today.
2) Grades vs Learning: Rote disconnected learning as we find in a lot of forced STEM programs is not helpful. Richard Feynman made this observation with the contrast between the Brazilin and US education system. I myself have seen it with Chinese and Indian education systems in STEM.
3) Mass vs 3-sigma: Is it better to be better at Mass STEM education or 3-sigma ? In general, technology invention is more like basketball. 3-sigma individuals outrank a mass of engineers. Finding, helping, and hoarding the worlds' best talent is very important. Also, mass "jobs" get designed out by technology over time by the 3-sigma solutions.
Finally, the most important point... freedom is really important. I don't mean freedom in some egalitarian nice-to-have sense. I mean freedom-of-thought in the problems to attack and the freedom-of-thought in the economy to consume goods/services (the plasticity of an economy).
The subtle consequence of all the judgement connected with grades[useless in my view] as well as the constraining thought process of the latest technological thrust (STEM as an example) has an impact on innovation (local and global). At the end, the rate of innovation is the only thing that really matters.
To enable the above, you need three things:
1) a service oriented education sector... as I said on-demand and early free.... [believe it or not..we are almost there...and I don't mean the establishment in education]
2) encouragement of citizens to solve problems around their passions [bottom-up innovation]
3) a rich "soil" of financial support for experimentation of ideas
Note.. for all of this to work, there is a severe need for basic values: trust, earnestness, hard-work..etc. This perhaps maybe the most important goal of the k-4 system... and of course parents. Part of the way the US works is because it has large elements of the above three.
Overall, it would be good not to destroy what works well in the zeal to follow some mass model.
Ii know you used the example of fab factories because they are shiny and high tech, but it is really a lot wider than that. I work for a chemical company. Our process is highly automated. The control computers can run the process better than a person could, because the computer doesn’t get bored. But while the computer is doing it’s thing, heat and dirt and acid are working to tear the equipment apart. We need skilled operators who can monitor the control graphs and go though the plant to monitor all of the equipment to handle the inevitable problems that the computers can’t handle. This takes basic math skills and plenty of critical thinking skills to do successfully. The manufacturing jobs that come back to the US in the future will be very different from the ones that moved away in the 80s and 90s.
One question I’d love an answer from opinion writers of pieces like this is:
“What level of math does the author believe the average American is capable of attaining?”
Because without an answer to that fundamental question, I’m always left wondering “why bother?” after reading pieces like this.
I’m currently convinced that the average American, heck, the average human, is bad at math (I most definitely include myself in this category!) and will not be capable of learning math to the level required for all of these jobs of the future.
Given the strong bifurcation of math ability required for jobs like those highlighted here, and most other occupations that require little more than middle school math, where is the value in raising the level of math attainment for the fat slice of the middle in the country?
A nation where everyone learns basic calculus is likely not going to be materially better off than one where everyone can do basic arithmetic so long as they both have enough people to fill their extremely advanced positions. That’s where the entirety of our focus should be, not on raising our median math education levels to a still unsatisfactory level.
I’ve never heard anyone argue for less STEM to increase egalitarianism -- citations would be great. We do need STEM but we need humanities just as much. Complete and well-rounded education is vital. Start by federally funding it so our southern citizens can get a decent education.
My youngest daughter was diagnosed as an Aspie in 2d grade. She was smart and literally hated middle school and its social games. 6 weeks into 6th grade she was still repeating math she knew (I had supplemented her math education to make sure she knew it), so I had her jumped to 7th grade math. In 8th grade I had her do 50% online education, took her to the high school (along with numerous other parents) for her math as the middle school didn't offer that level of algebra, and had her do her science, art/music, and physical education at her middle school. I had her skip 8th grade and do geometry by correspondence over the summer after 7th grade. She did Algebra 2 in 9th grade, precalc by correspondence over the next summer, and then calc for college credit in 10th grade. She dropped out of high school then and did early admissions to the state university, where she did her BS and MS in civil and structural engineering. I had to give the principals hell - they kept trying to get in the way. I told the principals that I did not care about state education requirements or school requirements - I was driving her education against college admission requirements - and high school graduation was not always required. She HATED the social scene in school, particularily in middle school.
My son wasn't quite as dilligent or mathematically talented, but he too did calculus in what would have been 11th grade - he did Running Start. and almost graduating with his associates. He did a Business - MIS degree in 7 quarters and then a MS in MIS - Data Security.
Too many of the Running Start students are slackers looking to escape the schedule of high school and then taking the easiest classes to graduate rather than stacking up transfer credits (and putting in the time to master the material) to prepare for the rest of their undergraduate education.
I am old and the kids knew that they didn't have backup - they had to be self supporting as soon as they graduated. We did not know if I would be forced to retire before they graduated - ageism in the tech industry is the rule, not the exception.