Math teachers should be more like football coaches

This does not happen in math either. What does happen is some students have significantly more preparation than others (including, for example, introspecting more while doing their homework for several years, playing board games, solving logic puzzles, building with construction toys, ...) and do more or better focused practice during a course.

Better-prepared students already know a significant part of the material being presented in a course, have extensively thought about ideas which are similar, or have improved their generic problem-solving skills. They end up stuck behind fewer blocking misconceptions.

"This book compares how traditional schools can, metaphorically, act like a ski school, which temporarily organizes its students into classes where each student shares a specific and common instructional need but has previously mastered the critical prerequisites that will enable new learning to take place."

https://www.amazon.com/General-Systems-Unique-Learning-Schoo...

This part is simply untrue. The problem with math education is largely one of poor communication between student and teacher. The biggest being that the teacher tends to be unable to sufficiently vary the way they instruct to adapt to the way a particular group of students in a class need to hear it. This is amplified by the fact that most math teachers tend to be frightened by the subject [1]. That fear tends to be passed on from generation to generation despite its irrationality. Thus it persists memetically. Add to this the fact that education has been trending towards standardized testing, and most incentives to create engaging ways of expressing these concepts are actively discouraged if not futile (within the system).

Mathematics isn’t something to be “signed up for” in the sense that it is pervasive in every single aspect of our lives. With this being stated, there is an important aside that must be made. People tend to focus on the relationship between the student and the teacher and the failure of communication there. However, there is an equally important chasm that remains hidden in plain sight—the relationship between educators and the academy itself. Mathematicians tend to take pride in performing their research for its own sake without concern for pragmatic applications. While there is merit to performing research without having to immediately justify its utility, that lack of justification should stop at practical applications to feed whatever social machinery is en vogue for the time period. It absolutely is not an excuse to put no effort into communicating what is being worked on in terms that others can understand. Thus we see a lack of empathy and communication from the genesis of mathematical ideas to the ones who find/create[2] those ideas to the ones who must pass that knowledge along to the ones who must absorb and apply this knowledge as students in order to expand their horizons or to simply find food, shelter, and partners for procreation.

[1] Calling Mathematics a subject is actually disingenuous since it encompasses so many different ways of thinking, conceptualizing, and expressing.

[2] Depending on your point of view, either concepts are “discovered” by the mathematician or they are “created” and did not exist before the creation.

So I think both sorts of dynamics exist in the university (though formal HS probably does little to no "coaching" type activity and math stars coach each other in high school math teams).

I don't think very many folks who were herded into the PE classes I've taken learned that much and everyone in the honors undergraduate seminar learned a lot. I can't prove that anyone who wants to can learn strong math but I don't think the results of a typical mass-level math class proves much about this either way. (I too, have taught your typical average crappy undergraduate math class).

Urschel's opinion piece points out how the two different systems treated him as an individual differently. He wonders (rightfully) what a world where the type of individual attention and energy that he was able to receive in football could be also received in mathematics. This is a very microscale discussion.

Your response (and many others) point out that on the macroscale, the two systems are incredibly different. But I feel that a lot of the macroscale differences aren't nessecary inherent. Like, they might be super baked in through path dependence, but they aren't intrinsic to the "platonic ideal" of producing football players and mathematicians.

Who's forcing anyone to do an undergraduate maths degree? And who does a degree in maths without being extremely motivated?

Learning the basics of snowboarding is the equivalent of learning how to count or do basic arithmetic. Most everyone is capable of counting and finds it immediately useful, as soon as they have money and see prices of things.

You're comparing snowboarding 101 to advanced math, of course they're different. Most people will not be able to do advanced snowboarding either, nor will they be motivated to do it.

And we're not even talking about the fact that they feed steroids to those at the top (220 pounds in high school--not bloody likely without some major chemical enhancement).

As I read it, the article is the author mentioning that as a student he had a poor idea of what professional mathematicians did, that he received a lot of encouragement from his athletic coaches but not math teachers, and finally wishing that as a kid he knew what excitement was possible from mathematics. So if I were picking a title for the article I might choose “I wish math students got the kind of motivation [from their teachers] that athletic students get [from their coaches]”. (This is wordy and poor as a headline of course, but it's probably a better reflection of the article.)

How much of the differences you mention affect these points, do you think? That might be more illuminating.

This is major difference from my experience. My math students are examined externally so I'm the one helping them achieve their goal.

My experience of teaching math is more similar to your experience of teaching snowboarding. Your last point drives this home - in my experience, anyone around average intelligence can ace school math if they have a coach who helps them approach it in the right way.

For the same hours of instruction block classes leads to poorer long term recall. Five students is the absolute upper limit of what can be considered tutoring. By the time you have ten students you’re into class teaching where the difference in learning between small and large class sizes exist but are not large enough to justify the greater expense in terms of learning.

> Snowboarding students have usually quite similar level of abilities. Of course, some learn faster than others but nobody improves 10 times faster than someone else (which happens in maths). If a student is better, she/he can move to another group the following week (something impossible in my math class).

We’ve known the solution to this problem since at minimum 1968 when Mastery based learning was named, and the first research on the topic dates from the 1920s. Don’t let students proceed to topic 2 until they’ve gotten 80 or 90% on a test on topic 1.

> As a math teacher, I'm also the one who evaluates them, and who stands between them and their degree. As a snowboarding coach, I'm the one who helps them achieving their goal. Different dynamic.

Again, this is a solved problem. AP tests exist. Math contests and the Putnam exist. The Royal Statistical Society used to set exams covering the first year, first two years and entire curriculum of a U.K. statistics degree, which latter was sufficient to get you onto a Master’s programme. They stopped due to lack of demand. Separating teaching and certification exams is not that hard.

Apart from differences in native ability Mastery learning and separating certification and teaching functions solves these problems. For whatever reason they’re not widely applied. In a developing nation context with very large classes the UK’s DfID has found that the most cost efficient method of improving education is teaching children by skill level, not age, which looks awfully like mastery based learning. There may be individual university departments that do this. I’m not aware of any. The only institution I know of that does Mastery based learning is Lambda School. It’s hard and requires complete institutional alignment. The closest universities come to it are weed out classes and prerequisites. An economics professor of my acquaintance started requiring calculus for his game theory classes, not because he uses calculus, but because it assured him that all students will be solid in algebra.

Looking at what institutions do instead of what they say their aims are they look much more like ranking institutions than teaching ones. Ebbinghaus’ forgetting curve research dates from 1885 and is an educational curiosity rather than a core part of instructional design. Most students forget most of what they’ve learned. If we wanted to teach instead of rank education would look enormously different from what we see in practice.

IMHO this is basically the answer: 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.

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), 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.

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

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.

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.

(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

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.

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.

Note: I'm aware college football coaches are fired if on-field results are below-par.

There's a reason we talk about amateur mathematician cranks and not amateur pianist cranks.

I have zero interest or aptitude in playing musical instruments. Any amount of time spent trying to teach me that is utterly wasted for all parties involved.

For mathematically gifted children, Prussian-style schooling is a net negative. The way to ensure equal access to math education isn't to funnel more resources into the system that tortures the love of math out of generations of children. Some kind of organized program that allows students to opt out of traditional schooling for mostly self-directed education seems like it'd work much better.

I agree wholeheartedly that opportunities need to be made more available to everyone:

- Roughly 80-85% of those who get their Ph.D.s in math are male. That's a depressing statistic, and we need to do better.

- Nearly all Americans who get their Ph.D.s in math are white. African-Americans and Latino/a Americans are underrepresented. Moreover, perhaps surprisingly, so are Asian-Americans. (There are many Asians in math, but usually they grew up in Asia and only came to the US for university or grad school.) Here, too, we must do better.

That said, your comment strikes me as rather nihilistic. Because opportunities aren't equally available to everyone, they shouldn't be made available to anyone?

Most people aren't like Urschel: most students won't go on to love math or excel in it, no matter how many opportunities they have. But some people will. Those people are spread across ethnicities, across genders, across socioeconomic groups, all over the world. We need to do a better job of finding and nurturing talent from diverse sources. And we need to make individual mentoring more widely available, not less.

Have I misunderstood you? Am I saying things you disagree with?

I had many excellent, engaging, exciting history teachers. My Latin teacher was quirky and annoying, but loved the subject and it rubbed off on us. My biology teacher made me think I wanted to go into hard sciences.

I barely remember my math teachers. And I was good at math.

I don't believe my experience is unique.

I really struggled with the rote path and it wasn't until I could see more of the "why" and where it applies to practical situations that it all started to click.