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For those debating reading the piece -- do it! John Urschel wrote it, and he's probably the most qualified to do so. I think his point is broadly correct: the

by majos 7y ago

For those debating reading the piece -- do it! John Urschel wrote it, and he's probably the most qualified to do so. I think his point is broadly correct: the way we teach football, and the way talented football players get mentored in high school, often puts mathematics to shame. Two responses come to mind: 1. The incentive structures are different for high school math teachers and high school football coaches. If a high school football coach produces an elite college (or pro!) prospect then colleges will pay more attention to the coach. Get enough attention and the high school coach might become a college assistant, a college head coach, etc. There is a very real personal incentive for high school coaches to produce good players. In contrast, a high school math teacher who produces great math undergrads will probably get student appreciation and little else. Maybe they'll get to work with the math olympiads or something, but it's not like a college will recruit them as tenure-track faculty. Most great high school math teachers are doing it for pretty pure reasons. Admirable, but hard to expect from rational people. 2. Another big problem is that there are very few high school math teachers who have a good idea of what mathematicians do, or have strong math backgrounds in general. Makes sense given that most people with strong math abilities can make much more money doing something that's not teaching. Plus, based on almost a decade around people who study, research, and teach math, it's a pretty small fraction of them who get excited about teaching math to people who, for the most part, aren't into it anyway. In contrast, outside of very small schools, most high school football coaches love football, even if they didn't necessarily play at a high level. These factors combine to make math teachers not like football coaches. I don't have a great solution to either problem. I do wish there was some better, more organized way for bored and retired math people to help teach middle and high schoolers. I think for many such people the desire is there, but they're not going to start a whole organization and overcome a bunch of administrative hurdles (which, apparently, are particularly offensive to many people who like math) to do it.

yodsanklai 7y ago
> a high school math teacher who produces great math undergrads I don't believe a high-school teacher can produce great math undergrads. Personal anecdote, when I was in high-school (very small town in Europe), one of my friend was very gifted at maths. He would spend his time in math classes drawing cartoon. The curriculum didn't fit his abilities. Today he's professor in an Ivy league university. He would have been a great math undergraduate no matter who his high school teacher was. EDIT: It's just an anecdote and of course I'm not saying teachers are useless, but I wouldn't rate teachers based on how many stars they've produced.
- majos 7y ago
  Some people are sufficiently talented that a mediocre high school education doesn't hurt them. I think more people have enough talent and latent interest to "go pro" in math or STEM but won't get there without a good teacher taking an interest. A good teacher is not always necessary, but sometimes it is.
Ntrails 7y ago
I have a vague sense that I would like to semi retire into teaching maths, but have a horrible suspicion that it would be infuriating and tiresome rather than fulfilling :-/
Alex3917 7y ago
> I don't have a great solution to either problem. IMHO this is basically the answer: https://www.amazon.com/General-Systems-Unique-Learning-Schools/dp/1683196309 https://www.amazon.com/General-Systems-Unique-Learning-Schoo... The idea is having different teachers for each topic rather than for each grade level, and letting kids move from one topic to the next once they master it. And having some statistical software to detect when kids are getting stuck and need outside intervention. This model is much better at teaching, produces kids who are much more engaged and self-confident about math, and is also much less expensive than the traditional model for teaching math.
- Pulcinella 7y ago
  Unfortunately in America people are heavily, heavily resistant to changing the model of education, and I say this as a teacher. You have parents who say they hate math and hated their math classes but will also get very angry when you try to teach their child in a different way than they were taught. Even suggestions like “maybe we could still have a rigid bell schedule but just turn off the very annoying, physical bells” is considered too radical. I do understand some of it. Parents are very worried about their child’s future and economic anxiety of their child being “left behind” so they would rather go with a model that they know. The problem is the model they know doesn’t work well or at all for many, if not most kids. They worry about their children not learning under an alternative mode when the current model already drives them to playing Fortnite in their phone all day during class.
ThrustVectoring 7y ago
I agree with the incentive bits. My elementary school had an optional after-school Math Club for 5th graders, but that was mostly around because my dad ran it and was really motivated to foster my childhood curiosity about math. It was a fantastic experience and a hell of a lot of fun, but I see little reason for anyone to organize it outside of wanting their kid to do well. Maybe an enterprising HOA officer could fund/run it to try to improve home values by getting recognition for their local school.
HarryHirsch 7y ago
Another big problem is that there are very few high school math teachers who have a good idea of what mathematicians do It's a massive problem in America, the undereducation of teachers. There is a lovely book, "Easy as π?" (https://www.springer.com/us/book/9780387985213 https://www.springer.com/us/book/9780387985213), it grew from an overview course in elementary mathematics for teachers of secondary-level math in Russia. The idea is to have the teachers know their math and the reasoning behind it. Try that in America, the students will complain to the administration that they learn too much, instead they will take all sorts of strange classes at their School of Education.
chongli 7y ago
As an undergrad in mathematics at a school known to be strong in the subject, and as a prospective teacher, this article and your reply have given me a lot of food for thought. In contrast to inspirational professional athletes who act as role models for kids, Urschel mentions Von Neumann as an example of someone math teachers never mention. To that I would reply that Von Neumann was a profound genius, far beyond the capabilities of most high school math teachers, before he'd ever set foot in a classroom [1]. To mention him to any kid in the hope that they may draw inspiration from him is risky. I think you're just as likely to discourage the child by reinforcing the stereotype that mathematics is a subject suited only to geniuses. At the same time, I do think that math teachers could work harder to inspire kids and help them realize their potential. That the incentive structure for great teachers is not there is something I don't have answer for. Perhaps we as a society need to find better ways of providing financial support to all of those whose economic activity produces net positive externalities. [1] https://en.wikipedia.org/wiki/John_von_Neumann#Child_prodigy https://en.wikipedia.org/wiki/John_von_Neumann#Child_prodigy
ken 7y ago
> or have strong math backgrounds in general. Makes sense given that most people with strong math abilities can make much more money doing something that's not teaching. That argument supposes that high school teachers ended up with that career by looking at all job prospects and picking the one with the best economic prospects. Based on my experience, exactly none of them did that, in any field. My sophomore English teacher used to work at an aerospace company, and my biology teacher studied to be a physician, for example. > Most great high school math teachers are doing it for pretty pure reasons. Admirable, but hard to expect from rational people. Exactly so. Until we can reliably identify success in a field (like football wins), and then make the economic incentives align to promote that success, we can't hope to increase the number of great people in a field by economic metrics.
- majos 7y ago
  > That argument supposes that high school teachers ended up with that career by looking at all job prospects and picking the one with the best economic prospects. Not really. I'm supposing that people choose careers based on some combination of "I like this" and "I like how this pays me". Someone who can complete a good undergrad math program can probably make six figures as a software developer after a few years. They'd have to really like teaching, or hate software development, for teaching to provide more total utility.
- sturgill 7y ago
  My dream is to be independently wealthy and become a high school math teacher. I love math, I enjoy teaching, and I’m pretty good at it (I spend a good deal of time tutoring various youth in our church group). Why am I not a math teacher? Because I make 5x what a teacher makes. And I don’t want to raise three kids on a teacher’s salary. But I dream of being a teacher. Help kids out for nine months and spend three months hiking and camping with the family. But until then, I’ll just make a very comfortable living doing quant marketing and building out Bayesian models... I enjoy my work enough that the opportunity cost is not worth the forgone revenue to teach. Not yet, any way...
- sethammons 7y ago
  The problem with a metric akin to football wins is that you can be cut from the football team. Required high school math classes don't have the option to remove poor performers. Worse, they seldom can remove disruptive students.
impendia 7y ago
>very few high school math teachers who have a good idea of what mathematicians do (Professional mathematician here) That's unfortunate, and it may be solvable. Indeed, professional mathematicians are usually the ones teaching prospective high school teachers during undergrad, so we are the ones to solve it. You have define "what mathematicians do" properly, though. Here is a recent paper that I (and two others) wrote: https://arxiv.org/pdf/1708.00044.pdf https://arxiv.org/pdf/1708.00044.pdf Nobody is going to understand that without a lot of specialized training. That shouldn't be the goal. But problem solving comes close. For example, the "riddler" problems on fivethirtyeight.com are fantastic. Here is an even better example: - Write down all the integers which can be written as a sum of two integer squares: 0, 1, 2, 4, 5, 8, 9, 10, 13, ... Go up to at least 200. - Look for patterns. At first, your descriptions will be vague. Keep refining them; eventually you get a nice exact description of which integers are on the list and which aren't. - Why are these patterns true? You've made a conjecture, can you prove it? - Are there similar such patterns with related questions? For example, integers which can be written as x^2 + 2y^2? As the sum of two cubes, or of three squares? What is the special sauce needed to make your pattern tick? Can you prove more cases, or a still more general theorem? This problem is accessible to anyone who knows basic high school algebra, but it requires a lot of effort (and, probably, in most cases, a lot of mentorship to help the student through it). We should try to do more of this kind of thing with our students: it is exactly what mathematicians do.
- dan_mctree 7y ago
  >This problem is accessible to anyone who knows basic high school algebra Are you sure? Spotting the pattern isn't easy and I imagine a lot of high schoolers would give up in frustration while going down wrong roads. A good portion probably wouldn't even write down the numbers without errors. And even if you get past that, most high schoolers don't know how to do/write proofs. I'd be shocked if more than 10% could actually complete all your steps. With challenging problem solving like that I'm afraid you're going to breed a lot of resentment.
  - impendia 7y ago
    I never said it was easy. In fact, in some sense, the problem is much harder than you might realize at first. This particular rabbit hole goes down very deep. I am not urging this family of problem on all high school students. Rather, on all prospective high school math teachers. They should be able to communicate the joy of problem solving and discovery to any of their students who show an interest and aptitude -- which means these teachers should have done some of this themselves.
joe_the_user 7y ago
OK, I read it. It seems pretty simplistic tbh. I too was talented in math and got fairly little encouragement from my math teachers. Despite this, I had the math bug hard and on my own initiative took college math classes up the advanced undergraduate level at the time. I went on to take a lot of math but not to become a mathematician. In a sense, the world of advanced academic subjects and the world of sports have a certain resemblance - there's room for only a handful at the top, especially now days. There are probably more tenured mathematicians in the world than there are professional sports stars but there aren't many of either and "feeders" tend to produce more of potential stars than can ever fully succeed. So coaches pushing everyone with athletic talent towards a professional sports track actually do severe disservice to lot of disadvantaged students. Someone going a long way in math will end up with more skills than the nearly successful athlete but thinking about it, if teachers are going to be encouraging, they should encourage a balance of learning and let the highly decide by themselves if they want to go the extreme. Which is to say HS math teachers should be different but probably not more like coaches. Another HS anecdote - another friend of mine, the son of an actual math professor (who probably had more raw talent than myself), came back from a Harvard interview with the comment - "I'm not going to be a mathematician. I found out that to be a mathematician you have to be always thinking, at the grocery store thinking, because all of the easy problems have been solved and to solve the hard problems you need to think 24/7". This isn't to say that math shouldn't be taught differently. What's needed imo is a curriculum where teachers evangelize math, and math-as-mental-discipline, for most students rather than math-as-needed for career or math for the most advanced.

by majos 7y ago

yodsanklai 7y ago

> a high school math teacher who produces great math undergrads I don't believe a high-school teacher can produce great math undergrads. Personal anecdote, when I was in high-school (very small town in Europe), one of my friend was very gifted at maths. He would spend his time in math classes drawing cartoon. The curriculum didn't fit his abilities. Today he's professor in an Ivy league university. He would have been a great math undergraduate no matter who his high school teacher was. EDIT: It's just an anecdote and of course I'm not saying teachers are useless, but I wouldn't rate teachers based on how many stars they've produced.

majos 7y ago
Some people are sufficiently talented that a mediocre high school education doesn't hurt them. I think more people have enough talent and latent interest to "go pro" in math or STEM but won't get there without a good teacher taking an interest. A good teacher is not always necessary, but sometimes it is.

Ntrails 7y ago

I have a vague sense that I would like to semi retire into teaching maths, but have a horrible suspicion that it would be infuriating and tiresome rather than fulfilling :-/

Alex3917 7y ago

> I don't have a great solution to either problem. IMHO this is basically the answer: https://www.amazon.com/General-Systems-Unique-Learning-Schools/dp/1683196309 https://www.amazon.com/General-Systems-Unique-Learning-Schoo... The idea is having different teachers for each topic rather than for each grade level, and letting kids move from one topic to the next once they master it. And having some statistical software to detect when kids are getting stuck and need outside intervention. This model is much better at teaching, produces kids who are much more engaged and self-confident about math, and is also much less expensive than the traditional model for teaching math.

Pulcinella 7y ago
Unfortunately in America people are heavily, heavily resistant to changing the model of education, and I say this as a teacher. You have parents who say they hate math and hated their math classes but will also get very angry when you try to teach their child in a different way than they were taught. Even suggestions like “maybe we could still have a rigid bell schedule but just turn off the very annoying, physical bells” is considered too radical. I do understand some of it. Parents are very worried about their child’s future and economic anxiety of their child being “left behind” so they would rather go with a model that they know. The problem is the model they know doesn’t work well or at all for many, if not most kids. They worry about their children not learning under an alternative mode when the current model already drives them to playing Fortnite in their phone all day during class.

ThrustVectoring 7y ago

I agree with the incentive bits. My elementary school had an optional after-school Math Club for 5th graders, but that was mostly around because my dad ran it and was really motivated to foster my childhood curiosity about math. It was a fantastic experience and a hell of a lot of fun, but I see little reason for anyone to organize it outside of wanting their kid to do well. Maybe an enterprising HOA officer could fund/run it to try to improve home values by getting recognition for their local school.

HarryHirsch 7y ago

Another big problem is that there are very few high school math teachers who have a good idea of what mathematicians do It's a massive problem in America, the undereducation of teachers. There is a lovely book, "Easy as π?" (https://www.springer.com/us/book/9780387985213 https://www.springer.com/us/book/9780387985213), it grew from an overview course in elementary mathematics for teachers of secondary-level math in Russia. The idea is to have the teachers know their math and the reasoning behind it. Try that in America, the students will complain to the administration that they learn too much, instead they will take all sorts of strange classes at their School of Education.

chongli 7y ago

As an undergrad in mathematics at a school known to be strong in the subject, and as a prospective teacher, this article and your reply have given me a lot of food for thought. In contrast to inspirational professional athletes who act as role models for kids, Urschel mentions Von Neumann as an example of someone math teachers never mention. To that I would reply that Von Neumann was a profound genius, far beyond the capabilities of most high school math teachers, before he'd ever set foot in a classroom [1]. To mention him to any kid in the hope that they may draw inspiration from him is risky. I think you're just as likely to discourage the child by reinforcing the stereotype that mathematics is a subject suited only to geniuses. At the same time, I do think that math teachers could work harder to inspire kids and help them realize their potential. That the incentive structure for great teachers is not there is something I don't have answer for. Perhaps we as a society need to find better ways of providing financial support to all of those whose economic activity produces net positive externalities. [1] https://en.wikipedia.org/wiki/John_von_Neumann#Child_prodigy https://en.wikipedia.org/wiki/John_von_Neumann#Child_prodigy

ken 7y ago

> or have strong math backgrounds in general. Makes sense given that most people with strong math abilities can make much more money doing something that's not teaching. That argument supposes that high school teachers ended up with that career by looking at all job prospects and picking the one with the best economic prospects. Based on my experience, exactly none of them did that, in any field. My sophomore English teacher used to work at an aerospace company, and my biology teacher studied to be a physician, for example. > Most great high school math teachers are doing it for pretty pure reasons. Admirable, but hard to expect from rational people. Exactly so. Until we can reliably identify success in a field (like football wins), and then make the economic incentives align to promote that success, we can't hope to increase the number of great people in a field by economic metrics.

majos 7y ago
> That argument supposes that high school teachers ended up with that career by looking at all job prospects and picking the one with the best economic prospects. Not really. I'm supposing that people choose careers based on some combination of "I like this" and "I like how this pays me". Someone who can complete a good undergrad math program can probably make six figures as a software developer after a few years. They'd have to really like teaching, or hate software development, for teaching to provide more total utility.
sturgill 7y ago
My dream is to be independently wealthy and become a high school math teacher. I love math, I enjoy teaching, and I’m pretty good at it (I spend a good deal of time tutoring various youth in our church group). Why am I not a math teacher? Because I make 5x what a teacher makes. And I don’t want to raise three kids on a teacher’s salary. But I dream of being a teacher. Help kids out for nine months and spend three months hiking and camping with the family. But until then, I’ll just make a very comfortable living doing quant marketing and building out Bayesian models... I enjoy my work enough that the opportunity cost is not worth the forgone revenue to teach. Not yet, any way...
sethammons 7y ago
The problem with a metric akin to football wins is that you can be cut from the football team. Required high school math classes don't have the option to remove poor performers. Worse, they seldom can remove disruptive students.

impendia 7y ago

>very few high school math teachers who have a good idea of what mathematicians do (Professional mathematician here) That's unfortunate, and it may be solvable. Indeed, professional mathematicians are usually the ones teaching prospective high school teachers during undergrad, so we are the ones to solve it. You have define "what mathematicians do" properly, though. Here is a recent paper that I (and two others) wrote: https://arxiv.org/pdf/1708.00044.pdf https://arxiv.org/pdf/1708.00044.pdf Nobody is going to understand that without a lot of specialized training. That shouldn't be the goal. But problem solving comes close. For example, the "riddler" problems on fivethirtyeight.com are fantastic. Here is an even better example: - Write down all the integers which can be written as a sum of two integer squares: 0, 1, 2, 4, 5, 8, 9, 10, 13, ... Go up to at least 200. - Look for patterns. At first, your descriptions will be vague. Keep refining them; eventually you get a nice exact description of which integers are on the list and which aren't. - Why are these patterns true? You've made a conjecture, can you prove it? - Are there similar such patterns with related questions? For example, integers which can be written as x^2 + 2y^2? As the sum of two cubes, or of three squares? What is the special sauce needed to make your pattern tick? Can you prove more cases, or a still more general theorem? This problem is accessible to anyone who knows basic high school algebra, but it requires a lot of effort (and, probably, in most cases, a lot of mentorship to help the student through it). We should try to do more of this kind of thing with our students: it is exactly what mathematicians do.

dan_mctree 7y ago
>This problem is accessible to anyone who knows basic high school algebra Are you sure? Spotting the pattern isn't easy and I imagine a lot of high schoolers would give up in frustration while going down wrong roads. A good portion probably wouldn't even write down the numbers without errors. And even if you get past that, most high schoolers don't know how to do/write proofs. I'd be shocked if more than 10% could actually complete all your steps. With challenging problem solving like that I'm afraid you're going to breed a lot of resentment.
- impendia 7y ago
  I never said it was easy. In fact, in some sense, the problem is much harder than you might realize at first. This particular rabbit hole goes down very deep. I am not urging this family of problem on all high school students. Rather, on all prospective high school math teachers. They should be able to communicate the joy of problem solving and discovery to any of their students who show an interest and aptitude -- which means these teachers should have done some of this themselves.

joe_the_user 7y ago

OK, I read it. It seems pretty simplistic tbh. I too was talented in math and got fairly little encouragement from my math teachers. Despite this, I had the math bug hard and on my own initiative took college math classes up the advanced undergraduate level at the time. I went on to take a lot of math but not to become a mathematician. In a sense, the world of advanced academic subjects and the world of sports have a certain resemblance - there's room for only a handful at the top, especially now days. There are probably more tenured mathematicians in the world than there are professional sports stars but there aren't many of either and "feeders" tend to produce more of potential stars than can ever fully succeed. So coaches pushing everyone with athletic talent towards a professional sports track actually do severe disservice to lot of disadvantaged students. Someone going a long way in math will end up with more skills than the nearly successful athlete but thinking about it, if teachers are going to be encouraging, they should encourage a balance of learning and let the highly decide by themselves if they want to go the extreme. Which is to say HS math teachers should be different but probably not more like coaches. Another HS anecdote - another friend of mine, the son of an actual math professor (who probably had more raw talent than myself), came back from a Harvard interview with the comment - "I'm not going to be a mathematician. I found out that to be a mathematician you have to be always thinking, at the grocery store thinking, because all of the easy problems have been solved and to solve the hard problems you need to think 24/7". This isn't to say that math shouldn't be taught differently. What's needed imo is a curriculum where teachers evangelize math, and math-as-mental-discipline, for most students rather than math-as-needed for career or math for the most advanced.