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 suspect that AI is going to be a major breakthrough in teaching.
Sadly, far too many teachers view it as evil, but being able to 'individualize' a lesson to a student based on how they learn, would be a major game changer.
For now, I really believe that we must get Americans, esp. those who have talent, to take up needed fields/trades. That is why I have been pushing an idea for some time.
I hope you are right but we aren't even using the technology we already have when it conflicts with tradition and status. I fear what will happen is that the schools won't use it so only the students who are motivated and wealthy enough to use it on their own time benefit.
Just because a number of teachers have chosen to NOT use the technology available, does not mean that AI will not be used. I used to teach Computer Science, swimming, as well as TA in Microbio. Students learn in very different ways.
An AI will quickly adjust to what a student needs and help them to see different approaches to solving issues.
I've seen a lot of that technology, and a lot of it is awful. I can't think how many students I've had to help deal with systems that can't handle answers like 1/2. Students, and their tutors, spend a ridiculous amount of time trying out 0.5, 1 / 2, one half and so on. I wish I were making this up.
Which is why the AI will make a huge difference. General applications do not have the means to adjust how a lesson/idea is conveyed. However, an AI designed for teaching will almost certainly be geared around teaching in recognizing how a student learns and adjusting to them, like a top teacher would do.
Most human tutors are terrible at this. It requires knowing the subject extremely well and being able to develop models of the student's misunderstanding. Liping Ma wrote a book on the problem of teaching arithmetic and pointed out that one big problem was that elementary school teachers don't understand it very well. One can teach arithmetic by dint of sheer practice and drill, but this knowledge is brittle and leaves important concepts unlearned, something that will bite later.
That's true. If you understand math well, you generally a) have better careers than teaching elementary school and b) are probably not interested in teaching young kids.
It doesn't help that public elementary school teachers are heavily unionized, which means that teachers are quite restricted in how they teach. If it's different, then either the administration or the union can block it if they don't understand it.
However, AI as it is currently implemented would be the opposite of a good tutor. It would be correct only by happenstance, and judging how a pupil made a mistake would also be accidental. We are impressed by the latest AIs mostly because it's hard to conceive of just how much human "communication" conveys the opposite of new knowledge, i.e. surprise.
The main problem is that modern LLM based AI doesn't have a model of how mathematics works or how students misunderstand it. Just relying on a corpus of tutoring sessions and examples isn't going to get you that. You might get something that helps a few students, but it's going to be about luck.
I run into this as well in my own teaching. My big problem is that in any given class there are students who:
(1) have no genuine interest in the material, are just there to get a passing grade (because someone in administration--not me--told them that they need to take calculus to graduate), and essentially demand that I just teach them an algorithm that they can faithfully execute to get the right answer each time, because heaven forbid that I ever ask them anything that requires demonstrating genuine understanding or problem solving skills by not being 100% isomorphic to a problem they’ve done before;
and (2) who have no interest in learning computational techniques unless they understand where they come from, why they work, what problems they solve, why these problems actually matter, and how they relate to other concepts they’ve learned.
Obviously as a mathematician I’m biased toward the latter group--if you can take a derivative by hand but can’t tell me what a derivative is in a few sentences then you’re not really learning anything useful that a computer can’t do more efficiently. But the former group is much larger typically.
What I’d really love is to just be able to teach two different versions of the same course and sort students perfectly into them.
Rote learning is way underrated in Western societies. Yes, conceptual understanding is very important. But at a basic level you have to be able to do the math, and it's ok if conceptual understanding doesn't come easily at first. For so many things I've learned I just kind of rote learned it first, then applying it made me understand what it was I was actually doing.
Yeah. This was my experience. All the years already spent trying to achieve what Noah is advocating mostly resulted in a system of teaching STEM that does a great job turning students off of math.
Many people don't like learning new things. The cognitive dissonance involved is an acquired taste. To make up for that often requires that the person realizes that learning this new technique will allow them to do something that is fairly immediately applicable.
School has increasingly distanced itself from teaching anything with immediate applicability, so you remove shop and home ec and add in calculus (and teach it badly because many of the high school teachers don't really understand it well).
And yes, having metrics for teachers is like having metrics anywhere else - the median teacher will do things that increase the metrics, regardless of whether actual teaching takes place. Although not having metrics gets us back to the underwater basketweaving courses of the 70s.
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 tend to think this is true of education in general. While having the "mental furniture" of facts and general concepts is definitely valuable in its own right, I still think education spends too much time on survey learning that most students will forget pretty fast anyway. Building deeper skills, whether in math, writing, or deep thought in other areas, seems to be more based on lots of motivated practice. I am neither a student nor a teacher, but it does seem like the educational system is slowly moving in this direction. I still think we have a ways to go.
I hate to be like "AI will solve this" but nah, AI will literally solve this. As far as I can tell the basic technology to build a really amazing guided-practice tutoring system for relatively simpler subjects already exists. Indeed just asking vanilla, consumer ChatGPT in words to act like this already gives you an amazingly good "first draft" system, although I'm sure if you actually tried to deploy it to real child learners it would be a disaster in any number of ways.
Come to think of it, remember ten years ago when people thought self-driving cars were a basically solved problem, but would be strangled in their grave by safetyist inattention to tradeoffs among different kinds of risk? That panic seems to have been unfounded, but it might well come true for a technology like an AI tutor that teaches you math. Honestly the profession is so screwy that I can imagine them trying to get it treated outright as cheating - doing your homework with a bot, basically - but even if that doesn't happen, they will absolutely slow-walk it and try to minimize its use because it's demeaning and unfun and feels like being replaced by a robot, even if that has nothing to do with the actual long-term lived experience of AI-driven transformations in the teaching profession as it shakes out.
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.
I agree with you, having a predisposition to mathematics, I grasped everything on the fly, no matter how I was taught the subject. But at the same time, my peers couldn't grasp topics that didn't seem complicated to me. And just for those kinds of people, better teaching would help raise the basic level of understanding.
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.
Now that’s the kind of burn I can get behind! “Left STEM, went to economics”. Economists are famous for their physics envy. They even believe they’re in STEM when everyone knows perfectly well they’re in Marketing.
That being said, there are more than enough superlatively excellent PhD material scientists and electrical engineers with exactly the right skills to fulfill the labor needs of fabs in California.
As someone with a physics background, I’m a little disappointed you didn’t seem to pick up on the difference between physics and engineering. Quantum mechanics is physics. Wafer fab is eng
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.
Land at Polaroid had a great way of getting top talent in chemistry. He hired from women's colleges. He'd find bright English majors who had good science grades and Polaroid would teach them chemistry. It was brilliant arbitrage. Of course, those women would be STEM majors at familiar schools today.
I'd be curious to hear what you need them to pick up and how fast, and how they aren't capable of picking it up fast enough. Are they fired, or do they muddle through? What's the impact on the organization?
I think natural ability plays a role, but I also think quality of education plays one. Maybe it's 50/50. For instance, I wonder how many of these people got the credential but didn't really learn the stuff well, thus putting them at a disadvantage for learning the new topics on top that you need.
1. Read a thousand page technical document over 4 days and be able to answer 8 out of 10 questions about the material that was covered.
2. A problem has come up that isn’t specifically covered in that document. But scattered throughout are bits of information that hint at a solution. Are they able to pull those bits of information together to solve a novel problem?
And yes they get fired.
I think 2 is especially difficult. You can drill people that A=B and a week later drill them thar B=C. Then six months later we’re out of B can we hook hook A directly to C? A lot of people don’t have the cognitive horsepower to remember those two facts and then make the logical leap that you can link A to C because C=B and B=A.
The Germans used to call that sitzfleisch, the sheer ability to sit down on one's ass and solve a problem even if it means reading through the damned documentation (RTFM) and figuring out the mindset of the system designers and enough of the system design to be able to make useful inferences. It can be massively painful at both ends after a while, but sometimes it is the only path.
It's not something an LLM can do because it involves building a model and making problem specific inferences.
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.
If this were true, it would mean that any group of students with approximately the same background knowledge in the same class & doing the same homework would learn about as well. Thus, for example, a group of students who have all learned algebra to a similar level & who are all in the same calculus class & doing the same homework would all learn calculus about as well. This seems so contrary to normal experience, as well as previous research, that it seems more likely that this study is somehow inaccurate.
Like all research, this study has strengths and weaknesses. 7000 thousand students and 200 data points each is a huge number for this kind of research. They got those numbers by using data from an online instruction program. The data is not broken down by subject, or student demographic. They report that they initially intended to identify factors that characterized high-performance learners but instead found that there is no such thing and that everyone learns at the same rate. You will, of course, apply your Bayesean priors to this report and come to your own conclusions. I find it quite convincing. "Some readers may object that near constant student learning rate unrealistically implies that everyone can master advanced level calculus or interpret abstract data. Indeed, not everyone has favorable learning conditions nor will everyone choose to engage in the substantial number of practice opportunities required. However, our results suggest that if a learner has access to favorable learning conditions and engages in the many needed opportunities, they will master advanced level calculus. "
That's an interesting paper. They use data from CMU's Datashop which is a database of automated course interactions, that is, the courses involved have been structured for computerized evaluation. The courses are divided into units and teaching is heavily practice oriented where each student is presented with a set of problems to solve, and the paper demonstrates that the difference in final progress strongly depends on the level of practice, that is, the amount of problem solving students performed.
There's nothing on the selection of students or more details on the structure of courses. I've taken courses like that and found that approach effective. On the other hand, I was usually was part of a selected group when I started. Would this be applicable in teaching elementary school mathematics? The US military applied this approach en masse during World War II, but they screened for promising candidates first.
I agree that the lack of detail on the selection of subjects and their demographics is a shortcoming of the paper. They do say that "elementary to college courses in math, science, and language" were observed. That suggests quite a broad range of learners and subjects.
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'm glad you weren't my high school algebra teacher.
My high school algebra teacher was very funny and used to tell us that if we didn't work hard, he would give us a "brain massage". This always sounded a little threatening. We never really knew what he meant, but it was enough to make us work hard. A few kids couldn't hack it in his class. But most could and most of us who stuck it out gained proficiency in grade ten algebra.
I was not enrolled in advanced classes. Most of the students in my high school took math to the eleventh grade. Most took at least one science class to eleventh grade. At the time, Calculus was not taught in high school in BC.
I then went to university in Ontario with a lot of students who had done high school in Ontario.
I'm not sure about the tracking in the Ontario school system. My suspicion is that it is set up to track a certain number of students into blue color jobs.
Compared to BC and Alberta, to me, Ontario feels both more elitist and more working class. There's a strange bifurcation there that doesn't really exist in BC and Alberta.
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.
Thanks for doing that for your daughter. Going through the miserable social side of school while also being bored to tears in most of your lessons is not a great preparation for adult life.
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 suspect that AI is going to be a major breakthrough in teaching.
Sadly, far too many teachers view it as evil, but being able to 'individualize' a lesson to a student based on how they learn, would be a major game changer.
For now, I really believe that we must get Americans, esp. those who have talent, to take up needed fields/trades. That is why I have been pushing an idea for some time.
https://windbourne.substack.com/p/how-to-fix-americas-higher-education
I hope you are right but we aren't even using the technology we already have when it conflicts with tradition and status. I fear what will happen is that the schools won't use it so only the students who are motivated and wealthy enough to use it on their own time benefit.
Just because a number of teachers have chosen to NOT use the technology available, does not mean that AI will not be used. I used to teach Computer Science, swimming, as well as TA in Microbio. Students learn in very different ways.
An AI will quickly adjust to what a student needs and help them to see different approaches to solving issues.
I've seen a lot of that technology, and a lot of it is awful. I can't think how many students I've had to help deal with systems that can't handle answers like 1/2. Students, and their tutors, spend a ridiculous amount of time trying out 0.5, 1 / 2, one half and so on. I wish I were making this up.
Which is why the AI will make a huge difference. General applications do not have the means to adjust how a lesson/idea is conveyed. However, an AI designed for teaching will almost certainly be geared around teaching in recognizing how a student learns and adjusting to them, like a top teacher would do.
Most human tutors are terrible at this. It requires knowing the subject extremely well and being able to develop models of the student's misunderstanding. Liping Ma wrote a book on the problem of teaching arithmetic and pointed out that one big problem was that elementary school teachers don't understand it very well. One can teach arithmetic by dint of sheer practice and drill, but this knowledge is brittle and leaves important concepts unlearned, something that will bite later.
That's true. If you understand math well, you generally a) have better careers than teaching elementary school and b) are probably not interested in teaching young kids.
It doesn't help that public elementary school teachers are heavily unionized, which means that teachers are quite restricted in how they teach. If it's different, then either the administration or the union can block it if they don't understand it.
However, AI as it is currently implemented would be the opposite of a good tutor. It would be correct only by happenstance, and judging how a pupil made a mistake would also be accidental. We are impressed by the latest AIs mostly because it's hard to conceive of just how much human "communication" conveys the opposite of new knowledge, i.e. surprise.
So spot on.
This is why I think that down the road, AI will do as good as a job as TOP teachers do today.
This will hopefully, solve a major issue which is levels of education depending on class.
The main problem is that modern LLM based AI doesn't have a model of how mathematics works or how students misunderstand it. Just relying on a corpus of tutoring sessions and examples isn't going to get you that. You might get something that helps a few students, but it's going to be about luck.
Initially. But that will quickly change.
I run into this as well in my own teaching. My big problem is that in any given class there are students who:
(1) have no genuine interest in the material, are just there to get a passing grade (because someone in administration--not me--told them that they need to take calculus to graduate), and essentially demand that I just teach them an algorithm that they can faithfully execute to get the right answer each time, because heaven forbid that I ever ask them anything that requires demonstrating genuine understanding or problem solving skills by not being 100% isomorphic to a problem they’ve done before;
and (2) who have no interest in learning computational techniques unless they understand where they come from, why they work, what problems they solve, why these problems actually matter, and how they relate to other concepts they’ve learned.
Obviously as a mathematician I’m biased toward the latter group--if you can take a derivative by hand but can’t tell me what a derivative is in a few sentences then you’re not really learning anything useful that a computer can’t do more efficiently. But the former group is much larger typically.
What I’d really love is to just be able to teach two different versions of the same course and sort students perfectly into them.
Rote learning is way underrated in Western societies. Yes, conceptual understanding is very important. But at a basic level you have to be able to do the math, and it's ok if conceptual understanding doesn't come easily at first. For so many things I've learned I just kind of rote learned it first, then applying it made me understand what it was I was actually doing.
Yeah. This was my experience. All the years already spent trying to achieve what Noah is advocating mostly resulted in a system of teaching STEM that does a great job turning students off of math.
Many people don't like learning new things. The cognitive dissonance involved is an acquired taste. To make up for that often requires that the person realizes that learning this new technique will allow them to do something that is fairly immediately applicable.
School has increasingly distanced itself from teaching anything with immediate applicability, so you remove shop and home ec and add in calculus (and teach it badly because many of the high school teachers don't really understand it well).
And yes, having metrics for teachers is like having metrics anywhere else - the median teacher will do things that increase the metrics, regardless of whether actual teaching takes place. Although not having metrics gets us back to the underwater basketweaving courses of the 70s.
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 tend to think this is true of education in general. While having the "mental furniture" of facts and general concepts is definitely valuable in its own right, I still think education spends too much time on survey learning that most students will forget pretty fast anyway. Building deeper skills, whether in math, writing, or deep thought in other areas, seems to be more based on lots of motivated practice. I am neither a student nor a teacher, but it does seem like the educational system is slowly moving in this direction. I still think we have a ways to go.
I hate to be like "AI will solve this" but nah, AI will literally solve this. As far as I can tell the basic technology to build a really amazing guided-practice tutoring system for relatively simpler subjects already exists. Indeed just asking vanilla, consumer ChatGPT in words to act like this already gives you an amazingly good "first draft" system, although I'm sure if you actually tried to deploy it to real child learners it would be a disaster in any number of ways.
Come to think of it, remember ten years ago when people thought self-driving cars were a basically solved problem, but would be strangled in their grave by safetyist inattention to tradeoffs among different kinds of risk? That panic seems to have been unfounded, but it might well come true for a technology like an AI tutor that teaches you math. Honestly the profession is so screwy that I can imagine them trying to get it treated outright as cheating - doing your homework with a bot, basically - but even if that doesn't happen, they will absolutely slow-walk it and try to minimize its use because it's demeaning and unfun and feels like being replaced by a robot, even if that has nothing to do with the actual long-term lived experience of AI-driven transformations in the teaching profession as it shakes out.
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.
I agree with you, having a predisposition to mathematics, I grasped everything on the fly, no matter how I was taught the subject. But at the same time, my peers couldn't grasp topics that didn't seem complicated to me. And just for those kinds of people, better teaching would help raise the basic level of understanding.
Personally part of the problem is society is to ambitious about what the baseline is meant to be.
For instance, Algebra has long been the standard. But society would be better off if everyone comprehended statistics instead.
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.
And really, why should it be put off as long as possible, because it would be much more useful to think backwards.
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.
Now that’s the kind of burn I can get behind! “Left STEM, went to economics”. Economists are famous for their physics envy. They even believe they’re in STEM when everyone knows perfectly well they’re in Marketing.
That being said, there are more than enough superlatively excellent PhD material scientists and electrical engineers with exactly the right skills to fulfill the labor needs of fabs in California.
Except if you get laid off:
https://www.msn.com/en-us/lifestyle/career/the-layoffs-are-here-for-those-who-chose-to-learn-to-code/ar-AA1ahACK
"there are still tons of openings."
source?
Wait, you have a physics degree?
Yes.
As someone with a physics background, I’m a little disappointed you didn’t seem to pick up on the difference between physics and engineering. Quantum mechanics is physics. Wafer fab is eng
Designing the machine is engineering.
Coming up with how it works on various aspects is chemistry and physics.
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.
Land at Polaroid had a great way of getting top talent in chemistry. He hired from women's colleges. He'd find bright English majors who had good science grades and Polaroid would teach them chemistry. It was brilliant arbitrage. Of course, those women would be STEM majors at familiar schools today.
I'd be curious to hear what you need them to pick up and how fast, and how they aren't capable of picking it up fast enough. Are they fired, or do they muddle through? What's the impact on the organization?
I think natural ability plays a role, but I also think quality of education plays one. Maybe it's 50/50. For instance, I wonder how many of these people got the credential but didn't really learn the stuff well, thus putting them at a disadvantage for learning the new topics on top that you need.
Two examples:
1. Read a thousand page technical document over 4 days and be able to answer 8 out of 10 questions about the material that was covered.
2. A problem has come up that isn’t specifically covered in that document. But scattered throughout are bits of information that hint at a solution. Are they able to pull those bits of information together to solve a novel problem?
And yes they get fired.
I think 2 is especially difficult. You can drill people that A=B and a week later drill them thar B=C. Then six months later we’re out of B can we hook hook A directly to C? A lot of people don’t have the cognitive horsepower to remember those two facts and then make the logical leap that you can link A to C because C=B and B=A.
I have never read a thousand page tech doc. And I promise I never will.
The Germans used to call that sitzfleisch, the sheer ability to sit down on one's ass and solve a problem even if it means reading through the damned documentation (RTFM) and figuring out the mindset of the system designers and enough of the system design to be able to make useful inferences. It can be massively painful at both ends after a while, but sometimes it is the only path.
It's not something an LLM can do because it involves building a model and making problem specific inferences.
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.
If this were true, it would mean that any group of students with approximately the same background knowledge in the same class & doing the same homework would learn about as well. Thus, for example, a group of students who have all learned algebra to a similar level & who are all in the same calculus class & doing the same homework would all learn calculus about as well. This seems so contrary to normal experience, as well as previous research, that it seems more likely that this study is somehow inaccurate.
Like all research, this study has strengths and weaknesses. 7000 thousand students and 200 data points each is a huge number for this kind of research. They got those numbers by using data from an online instruction program. The data is not broken down by subject, or student demographic. They report that they initially intended to identify factors that characterized high-performance learners but instead found that there is no such thing and that everyone learns at the same rate. You will, of course, apply your Bayesean priors to this report and come to your own conclusions. I find it quite convincing. "Some readers may object that near constant student learning rate unrealistically implies that everyone can master advanced level calculus or interpret abstract data. Indeed, not everyone has favorable learning conditions nor will everyone choose to engage in the substantial number of practice opportunities required. However, our results suggest that if a learner has access to favorable learning conditions and engages in the many needed opportunities, they will master advanced level calculus. "
That's an interesting paper. They use data from CMU's Datashop which is a database of automated course interactions, that is, the courses involved have been structured for computerized evaluation. The courses are divided into units and teaching is heavily practice oriented where each student is presented with a set of problems to solve, and the paper demonstrates that the difference in final progress strongly depends on the level of practice, that is, the amount of problem solving students performed.
There's nothing on the selection of students or more details on the structure of courses. I've taken courses like that and found that approach effective. On the other hand, I was usually was part of a selected group when I started. Would this be applicable in teaching elementary school mathematics? The US military applied this approach en masse during World War II, but they screened for promising candidates first.
I agree that the lack of detail on the selection of subjects and their demographics is a shortcoming of the paper. They do say that "elementary to college courses in math, science, and language" were observed. That suggests quite a broad range of learners and subjects.
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.
No. The average human is likely capable of high school level geometry, algebra and trigonometry.
I'm glad you weren't my high school algebra teacher.
My high school algebra teacher was very funny and used to tell us that if we didn't work hard, he would give us a "brain massage". This always sounded a little threatening. We never really knew what he meant, but it was enough to make us work hard. A few kids couldn't hack it in his class. But most could and most of us who stuck it out gained proficiency in grade ten algebra.
I was not enrolled in advanced classes. Most of the students in my high school took math to the eleventh grade. Most took at least one science class to eleventh grade. At the time, Calculus was not taught in high school in BC.
I then went to university in Ontario with a lot of students who had done high school in Ontario.
I'm not sure about the tracking in the Ontario school system. My suspicion is that it is set up to track a certain number of students into blue color jobs.
Compared to BC and Alberta, to me, Ontario feels both more elitist and more working class. There's a strange bifurcation there that doesn't really exist in BC and Alberta.
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.
Thanks for doing that for your daughter. Going through the miserable social side of school while also being bored to tears in most of your lessons is not a great preparation for adult life.