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I used to teach maths (undergraduate level) and I was also a snowboarding coach. These two roles aren't really comparable.

- In my snowboarding groups, there were usually about five students, no more than ten. We would spend a full day together, and I had time to spend quite a lot of time with each student individually. In my math "exercise" classes, there were up to 25 students. I would see them 3 hours a week (they had other teachers as well).

- Unlike math students, snowboarding students were all very motivated. Nobody forced them to be there.

- 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).

- 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.

- I'm confident that anyone without disability can learn to be a decent snowboarder after a few weeks of training. But I think that a lot of students don't have what it takes to learn the mathematics curriculum they signed up for, no matter how you teach them.

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I think these types of reflection (both in the original opinion piece and your response) are useful because they highlight two very different perspectives to the situation.

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.

>macroscale differences aren't [necessarily] inherent.

That's true up to a point. In general, phys ed classes/requirements in high school and university aside, if you're a thoroughly so-so team athlete in general, society is mostly perfectly fine with programs that strongly prioritize attention on the gifted.

To some degree this happens with academics at the university level. But in high school there's an, if not zero sum something close to, tradeoff between focusing on the students who are more motivated and/or have seemingly more aptitude for certain subjects vs. focusing resources on mainstream or even struggling students.

Whatever K12 math education is aimed at, I don't think it can possibly be "producing mathematicians." Producing human computers, maybe.

I've long suspected that K-12 math was designed as vocational training for the job "computer", which was a common job in the defense industry prior to its replacement by electromechanical and digital computers. For example, the computation of the firing tape measures for US artillery in WW2 would require enormous, precise calculations of just the sort that one practices in K-12 mathematics.

K12 Mathematics education is mostly a giant waste of time, teaching children things they’re just barely capable of learning now that they can pick up in a fraction of the time a few years later. Sudbury and other democratic schools can teach the entire primary school math curriculum to 12 year olds in around 40 hours. You can teach a nine year old to read to grade level in about the same amount of time. Primary school education works wonders at teaching children to sit still, do what they’re told and doing complex, boring tasks on command but as education it leaves a huge amount to be desired.

Of course the paradox to that is the fact that even though he had quite different attention placed on him in these two domains, he excelled at the highest level at both. Does that bring into question just how big an impact that experience really has?

I'm a coordinator at a K-12 school and I actually think about this quite a bit, and I'm not really sure I've made much progress. I've had the privileges of being an administrator in schools in different continents, serving vastly different populations, organized very differently, etc. And it has always shocked me how seriously we take some decisions, or how many hills staff is willing to die on for a sincere belief that deciding something one way or another is crucial to our work, yet knowing in the back of my mind another place where the approach was wholly different and yet the outcomes looked much the same.

> nobody improves 10 times faster than someone else (which happens in maths).

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.

And the effect is they improve 10 times faster than others who don't do all that.

Solving problems faster or comprehending a concept similar to one you already know is not the same as “improving 10 times faster”. Most of the “improvement” is already in the past.

In the same way a concert pianist doesn’t “improve 100x faster” than me if he can learn a new song in 6 minutes that takes me 10 hours to learn.

In the same way Bill Gates’s savings account doesn’t “improve 1 million times faster” than mine when we both earn interest.

The analogy for snowboarding might be taking an experienced snowboarder and a novice, and introducing some new type of obstacle on the slope. The experienced snowboarder will learn how to deal with it much faster than the novice.

> Solving problems faster or comprehending a concept similar to one you already know is not the same as “improving 10 times faster”. Most of the “improvement” is already in the past.

Continuing with the original example — these would be the newbie snowboarders who have prior experience with other board sports.

are you suggesting talents does not exist or only that it accounts for much less than factor of ten differences? what expertise or what evidence do you have supporting this belief?

There are definitely differences between people (e.g. differences in personality, interests, working memory, focus, emotional control, ...)

But my impression is that differences in (a) preparation/trained skills and (b) other context like diet, exercise, amount of sleep, family/relationship stress, illness, .... are more significant.

At the least, nearly everyone is capable of successfully handling the standard school curriculum, given a few hours per week of 1:1 tutoring by an expert.

> nobody improves 10 times faster

> At the least, nearly everyone is capable of successfully handling the standard school curriculum, given a few hours per week of 1:1 tutoring by an expert.

Your second statement does not imply the first one. "Nearly everyone" may be capable of successfully handling the standard curriculum, and still, given the same resources, and hours/week some people may progress 10x faster than others.

I am in starkest disagreement with this claim from the top-level comment:

> I think that a lot of students don't have what it takes to learn the mathematics curriculum they signed up for, no matter how you teach them.

Unless “a lot of students” means “the < 10% of the population with a severe mental handicap most of whom never went to college”... which okay, I am happy to concede that.

I also dispute that with all else held equal, certain people are inherently “10 times faster” at “improving” at mathematics. This just doesn’t seem plausible to me. For context, I have observed close-up several of the smartest math students in the world, and also spent a decent amount of time trying to help people who were heavily struggling with remedial coursework.

I would even claim that the latter group were “improving faster” than the former group.

Every trait or ability should be expected to more or less follow a normal distribution. There will be some few people at the extremes of the distribution. Most people will tend to the mean. For every trait.

>This does not happen in math either. What does happen is some students have significantly more preparation than others

Are you saying people don't have inherent different abilities in mathematics? That's hard to believe based on real-world examples.

On the low end of abilities scale, you have people suffering from dyscalculia[1]. You could repeatedly attempt to teach some of them algebra and calculus for years and they'd never be able to master it. It doesn't mean they're stupid; It's just that math in particular is very difficult for them. There are 20-year olds in college that have been subjected to math education for decades that still can't pass a remedial algebra class.

In contrast, on the high end, you have prodigies like Terrence Tao[2] that learned calculus at age 7. His brain literally burns less calories and requires less study time than most others to learn calculus.

[1] https://en.wikipedia.org/wiki/Dyscalculia#Common_symptoms

[2] https://en.wikipedia.org/wiki/Terence_Tao

Some people are missing a leg and can’t do snowboarding. But that’s not my understanding of what the previous comment was talking about.

Terry Tao is comparable to an Olympic athlete w/r/t preparation and focus. He has spent a dramatically larger proportion of his waking hours doing serious mathematics preparation than any non-mathematician alive, with mentoring from world-class experts. It is very hard to judge the relative “speed of improvement” vs. a person failing a poorly taught remedial algebra course in college (or whatever).

>Some people are missing a leg and can’t do snowboarding. But that’s not my understanding of what the previous comment was talking about.

I don't see why they would be excluded.

Before your reply, I wasn't sure if you'd argue that dyscalculia was not a real problem and that it was really a misdiagnoses based on bad teaching. In other words, a teacher with a emphatic belief in the power of "math tutoring" would claim to resolve all those math difficulties. It seems you seem to concede that dyscalculia might be a real cognitive issue so building our conversation on that...

Most probably suffering from it aren't even aware of that term "dyscalculia" to self-label themselves and instead would just say they are "bad at math". Doesn't it seem plausible that some would also be in those math classes and having trouble? They would be the ones that are progressing 10x slower than their peers.

If we accept that dyscalculia is real, and also accept another level of math ability such as autistic savants with lightning speed at calculating square roots or identifying 6-digit prime numbers, is another real but unteachable phenomenon, why can't there be a 10x difference in math ability for everybody else that's not explained by hours of effort spent?

>Terry Tao [...], with mentoring from world-class experts. It is very hard to judge the relative “speed of improvement” vs. a person failing a poorly taught remedial algebra course in college.

I find it interesting in your choice of qualifying the scenarios. E.g. Your sentences didn't also lay out the opposite scenario that 7-year old Tao learned advanced mathematics quickly in spite of bad teachers and the 18-year old student failed algebra even though he had a few good math teachers between ages 6 and 18 with ~5000+ hours of math schooling.

I know you're a big believer in the effectiveness 1-on-1 tutoring to teach math and that's great -- but -- that still doesn't explain the different abilities because some students learn faster without requiring extra 1-on-1 tutoring.

Something else has to explain the vastly different math abilities even when you statistically control for hours study, personal tutoring, etc.

Let's put it another way by comparing skills such as driving a car vs differential & integral calculus:

We can take a set of average adults that don't know how to drive a car and after a few hours of instruction and practice, ~99% will be able to drive a car well enough to navigate city streets.

If Tao's calculus ability by age-7 is not an exceptional math brain ability but really an example of an "average brain" with world-class "math tutoring", it means we should be able to devise a teaching method that can get 99% of average children to master calculus by age 7. Do you think this is possible? I seriously doubt any talented math instructor (Khan academy, 3Blue1Brown, etc) can guarantee they can structure a class to get 99% of 7-year olds to master calculus. Probably less than 1% of 7-year olds would be able to learn calculus. As an easier goal, it seems we can't even devise a math teaching method to get 99% of 17-year old teens to master calculus. I'd love to be wrong about this.

When I was younger, I tutored a bunch of people who claimed to be “bad at math”. None of them had any inherent cognitive problem that I could discern. In every case the issue was a combination of lack of preparation and severe anxiety caused by the high pressure and feeling inadequate, with the students never letting themselves relax enough to pay attention or think clearly.

If placed in a context for a few years where they were approaching math problems with open-minded curiosity, without the threat of judgment from a teacher, embarrassment/shame in front of a class, or poor scores with implied dire future consequences at getting answers wrong, I think any of them would have been just fine (not winning the IMO, but succeeding at class).

Obviously this is not true for everyone (some people e.g. are profoundly mentally disabled, or have some mental block around all sorts of symbolic reasoning), but cultural / pedagogical barriers to many people’s mathematical understanding are huge.

We can witness that there are some schools and societies that systematically do a better job of preparing their students than others. Without their students spending any more hours studying, or having some inherently different aptitude.

* * *

Driving a car just isn’t that hard. (Albeit quite dangerous.) If all you cared about was teaching 3rd grade arithmetic, and you were teaching 16 year olds, it would be not hard to cover the up-to-3rd-grade curriculum in a few dozen hours of practice.

>, and you were teaching 16 year olds, it would be not hard to cover the up-to-3rd-grade curriculum

In this conversation of hypotheticals, I don't see why we have to dumb down the goal from 7-year olds learning advanced math like calculus to the easier goal of 16-year olds learning 3rd-grade arithmetic.

Let's recap the discussion up to this point:

- You contend that some people don't improve at math 10x than others. (The 10x can be explained away by mostly fear, grades anxiety, culture, teaching, 1-on-1 teaching, fostering curiosity, etc)

- I mention 2 examples of different math ability: (1) lower math ability (maybe ~0.1x math progress) and (2) high math ability like Terence Tao (maybe 10x math progress)

Based on your comments [0], I think you believe my examples of varying math learning speed is invalid and that TT is an "average math brain" that benefited from great teaching. Ok, if that is true, why can't we make breakthrough math course that can teach 7-year olds to master calculus? Not just simple 3rd-grade arithmetic but advanced calculus.

If a math teacher believes strongly in the thesis that people like TT just have an average math learning speed (which means the general population is equal to TT in math learning ability), he could make a "Calculus for 7-year olds" course that could make more money than the "Baby Einstein" DVDs. Instead of graduates with math degrees accepting low-paying adjunct professor contracts for $30k/year, they can charge $50k to tutor/mentor the children of millionaires/billionaires and guarantee they can get a SAT MATH 760 score like 9-year old Terrence Tao.

Those breakthrough teaching courses haven't happened which means we still have a misunderstanding/overestimation about math pedagogy that doesn't explain TT's advanced math skills at his young age.

Also as a side note about fear and anxiety in math teaching... there are lots of horror stories from China, Japan, S Korea of kids getting physically punished (both by teachers and parents) for getting wrong math answers and yet they still become very good at math.

[0] >I also dispute that with all else held equal, certain people are inherently “10 times faster” at “improving” at mathematics.

Terry Tao was passionately interested in mathematics starting from an early age, had extreme amounts of support from experts, and spent incredible amounts of time and focused effort on it.

There are vanishingly few people in the world who spent as much time or had as much guidance at a similar age. It’s basically impossible to find a reasonable comparison vs. Tao of someone who had similar amounts of support and spent a similar amount of time and yet was never able to learn anything.

Humans (including young kids) are incredibly capable if they consistently practice something every day over the course of years, with useful feedback along the way.

Take for example the Polgar sisters, who became 3 of the strongest female chess players of all time because their family made chess success a long-term full-time family project. There is no particular reason to believe that any average baby adopted into the same family at birth would not also have become a skilled chess player.

Talking about passionate interest just passes the buck. Sure Tao was, but why was he passionately interested and can we replicate it in an average person?

>There are vanishingly few people in the world who spent as much time or had as much guidance at a similar age. It’s basically impossible to find a reasonable comparison vs. Tao of someone who had similar amounts of support and spent a similar amount of time and yet was never able to learn anything.

Sure. But do you think you (or someone with suitable skills) can replicate Tao's success? I don't think you can.

>There is no particular reason to believe that any average baby adopted into the same family at birth would not also have become a skilled chess player.

No reason, except that this sort of thing is very very rarely done despite the obvious rewards. Maybe most people wouldn't be inclined to go through such an arduous project, but authoritarian governments certainly will (and the Olympics prove that they do, and that this works to some extent, although not as spectacularly as you're selling it, for some sports). Where is the horde of Soviet Nobel laureates? Where are the North Korean Fields medalists?

> why was he passionately interested and can we replicate it in an average person?

That is a very good question. I would imagine there been a lot written about this, but I’m not sure how systematically it has been studied.

One thing that definitely does not work for the general student is typical classrooms with scores, grades, a fixed pace, and tons of bureaucratic overhead.

> this sort of thing [the Polgar experiment] is very very rarely done despite the obvious rewards

The “reward” to being a child prodigy isn’t that great. If the cost is most of the parents’ full-time attention, then most families can’t afford it. This doesn’t scale.

It does if you're a totalitarian state that really really wants to win an ideological battle by producing Olympic medals and Nobel prizes. Only one of those were produced.

Nobels are given by committees, so they are not necessarily a fair comparison.

Olympic medals are more comparable to IMO medals, and the latter were won disproportionately by communists states with dedicated preparation: China in the last 30 years, before that the Soviets, the Hungarians, Romanians and East Germans. [1]

[1] https://en.wikipedia.org/wiki/International_Mathematical_Oly...

I think the vast majority of math students don't fit the extremes you've mentioned. Personally I find it hard to blame anything but the system, at least in US schools. I began school in 2 Commonwealth countries and was a very average student, underperforming compared to my siblings. I moved to the US in 7th grade and was considered a genius, always getting in "gifted programs", etc. I was in possession of math books from my siblings high school and enjoyed them - ultimately finding that my bachelor's degree from an engineering school still didn't require math classes that got close to those books. So I have a hard time saying it's inherent ability holding math classes back.

>So I have a hard time saying it's inherent ability holding math classes back.

I wasn't saying anything about holding math classes back. My reply to jacobolus was specifically about the 10x difference in math skills progression and exploring what the underlying reasons could be.

What are some topics which were covered in your siblings' high school math textbooks which weren't required for your degree from a university? (You say "from an engineering school", but it's not clear if it was an engineering degree.)

Degree was computer engineering - less calculus than mechanical engineering, but all degrees from that department had a base standard of some 200-level courses from the math dept. for science & engineering majors. Linear algebra was probably most lacking - as it only covered system of equations and basic matrix operations and properties. Eigenvectors, subspaces, etc. I've had to learn all that myself. Calculus never actually exceeded AP calculus as taught in a US high school. Statistics included a pretty similar set of concepts, though the problems were much simpler (fewer layers of conditional probability) and didn't cover quite as many example of tests & methods, etc.

> Are you saying people don't have inherent different abilities in mathematics? That's hard to believe based on real-world examples.

Are you saying people don't have inherent different abilities in snowboarding ? That's hard to believe based on real-world examples (like Olympic champions)…

I believe that the argument was not against there being different abilities but about there being a much closer correlation between practice time and success in snowboarding than in mathematics.

I will note that (Soviet vs. US/European success) in (Olympic sports vs. Nobel/Fields medals) strongly supports this argument. Prodigy manufacture has been tried and shown to sometimes work in some fields.

The soviet success is a good example, but I don't think it supports your point: the soviets where very successful at training athletes to become gold medalists but they were also really successful at training mathematicians and physicists, even after losing almost all the educated class during the revolution and the following war (a majority a them were “White” Russians who fled the country).

You could argue that they were more successful at training athletes than scientists, because they have more gold medals than Nobel prize. But I don't think it really makes sense to compare those figures: soviets were doping there athletes a lot, and many of these gold medals comes from what I would call “mere performance sport” (like Athletics or Weightlifting), and they were competing with the West on the prestige given by gold medal, while the Nobel prize weren't a goal in itself. They were the first to reach space thought ;).

Also, you shouldn't dismiss the existence of an American “prodigy manufacture”: the American school and university system was extremely successful during the second third of the century and that's where most the American scientific success comes from, and the American sport system has also been excellent for a while.

The Soviets dominated the Olympics, but they produced 25 Nobel Prizes to America's 375. I might agree with you were the numbers less extreme.

Re: American prodigy manufacture, you have a good point and I should think about this more.

If I had to guess, you're a skeptic of the existence of IQ, or at least you're expressing a viewpoint in line with that. I don't know how to break it to you, but the mountain of evidence is not on your side.

Set aside Terence Tao and consider another mathematician, John Von Neumann. He had mastered differential and integral calculus, could converse in ancient Greek (among several other languages), and read through the entire 46-volume Allgemeine Geschichte in Einzeldarstellungen (essentially the history of the world, according to Wilhelm Oncken) at the age of 8, all without ever setting foot in a classroom. You can't seriously suggest that he benefitted from better teaching when he was entirely self-taught.

Contrast this with students who get all the way through high school and still cannot read or write above an elementary school level, who cannot perform basic arithmetic without a calculator, all of this despite thousands of hours of effort.

Von Neumann had the full-time 1:1 attention of a (presumably brilliant) nanny and significant 1:1 time from several additional expert private tutors starting from an early age, as well as a ton of parental support. That is way more useful than sitting in any classroom. Then he was the kind of kid who liked to walk around town doing arithmetic calculations in his head, and was passionately curious about everything and spent tons of time reading throughout his childhood. By the time he got to school he was several years more prepared than most of the other students.

The dude was obviously very clever, but it’s not like you can learn to speak several languages and understand advanced mathematics from scratch just by staring at the wall with no input or feedback.

My 2-year-old has a bigger vocabulary and better grammar than any kids his age at the local playground. That’s because we read challenging books together 2+ hours per day, not because he is some kind of superhuman genius. He also runs faster and with more stamina – because he gets a ton more practice than most running and playing outside. On the other hand, he doesn’t fluently speak 4 languages like the kids whose parents speak 4 languages to them daily, he isn’t so good yet at climbing or throwing or drawing since he isn’t very excited about those skills/activities yet and doesn’t practice them regularly. It’s not a priority for me, but I would not be at all surprised if he ends up learning basic calculus before age 10.

You're getting the story backwards. Von Neumann didn't start working with private tutors until after he had manifested his prodigious mathematical talent, in high school. He was a genius who could multiply and divide pairs of 8 digit numbers in his head, as a six-year-old.

The advanced mathematics he taught himself from textbooks. Sure, his parents provided him with the books, but he did the work on his own.

advanced mathematics from scratch just by staring at the wall

That's an absurd strawman.

From what I remember he was working with private language tutors starting as a toddler. I might not have that right though. I should read a decent biography sometime.

Just another tidbit: Von Neumann had no education in chemistry until adulthood, yet he was able to complete his degree in chemical engineering while simultaneously obtaining his PhD in mathematics, all in the span of 3 years.

This sounds amazing because it shifts blame for poor performance from lack of simplistic analogy to easier examples to a lack of preparation, but I don't think what you're saying is true. Some people are born with a higher ability for some things, I genuinely doubt that people in math majors not as quick on the uptake can just be brushed aside because they didn't "prepare" as well as they should have.

This is as absurd as saying that neurotypical children have more preparation for social interaction or better preparation than autistic children, or that Kalenjin children have more preparation for running than Khoisan ones. People vary enormously in their natural talents socially, academically and athletically.

https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss

> Gauss was a child prodigy. In his memorial on Gauss, Wolfgang Sartorius von Waltershausen says that when Gauss was barely three years old he corrected a math error his father made; and that when he was seven, he confidently solved an arithmetic series problem faster than anyone else in his class of 100 students.[9] Many versions of this story have been retold since that time with various details regarding what the series was – the most frequent being the classical problem of adding all the integers from 1 to 100.

https://en.wikipedia.org/wiki/Terence_Tao

Tao exhibited extraordinary mathematical abilities from an early age, attending university-level mathematics courses at the age of 9. He and Lenhard Ng are the only two children in the history of the Johns Hopkins' Study of Exceptional Talent program to have achieved a score of 700 or greater on the SAT math section while just nine years old; Tao scored a 760.[9] Tao was the youngest participant to date in the International Mathematical Olympiad, first competing at the age of ten; in 1986, 1987, and 1988, he won a bronze, silver, and gold medal. He remains the youngest winner of each of the three medals in the Olympiad's history, winning the gold medal shortly after his thirteenth birthday.

This comment seems a reply to the headline more than to the article. And given that headlines are almost never chosen by the authors of articles, it may be worth coming up with an alternative headline to the article.

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.

Not sure if you saw the book I linked to in my other comment, but that's the premise:

"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...

> I'm also the one who evaluates them... different dynamic.

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.

As a PE student, I have only taken large classes where the teachers barely gave any attention. I did, however, take an undergraduate honors seminar where there were seven self-selected students all of whom were there voluntarily and who got individual attention from the professor.

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).

The entire point of the article is to make them more comparable.

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.

But I think that a lot of students don't have what it takes to learn the mathematics curriculum they signed up for, no matter how you teach them.

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.

>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.

I'm sure if you spend enough resources on teaching a student to do something then almost every student will eventually get it, but at what point does the amount of resources spent become too much? Just the fact that math classes have 25-40 students each already means the resources we spend are limited and within those constraints it might very well be likely that those kids can't be taught this. It's not like we're suddenly going to get a massive surge in math teachers.

>Mathematics isn’t something to be “signed up for” in the sense that it is pervasive in every single aspect of our lives.

It might be pervasive, but most people could go through their life without learning half the mathematics that they do in school. I think studying mathematics is important to learn ways of thinking, but I also recognize that, in practice, most people have little need for most mathematics they learn.

> It might be pervasive, but most people could go through their life without learning half the mathematics that they do in school.

Most people do go through their life without learning half the mathematics they were notionally supposed to learn in school.

We had a school where math was the focus. So half of the whole time was dedicated to the math classes. And students were motivated because they've come there for exactly that.

I've always thought it would be better if teaching and evaluation were done by separate people.

In what respect. During teaching you constantly have to evaluate your students to establish feedback so you can adjust your teaching. Evaluation is an inherent part of teaching.

IME high-school teachers (my parents, wifes parents, and all my parents friends are teachers) could always predict the grade a student would get -- though sometimes public predictions were given differently as a motivator "you need to put a bit more effort in to get a C [minimum pass many employers used to look for]".

Maybe if you're seeking objectivity?

That's what standardized testing is. It has a bad failure mode where the curriculum starts to cater to the evaluator instead of the teacher's pedagogical needs.

> In my snowboarding groups, there were usually about five students, no more than ten. We would spend a full day together, and I had time to spend quite a lot of time with each student individually. In my math "exercise" classes, there were up to 25 students. I would see them 3 hours a week (they had other teachers as well).

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.

In addition, coaches "cut" those who don't make the grade. This allows them to focus attention on those who already have some aptitude.

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).

> Unlike math students, snowboarding students were all very motivated. Nobody forced them to be there.

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

> Who's forcing anyone to do an undergraduate maths degree?

People are forced to do maths classes as part of other degree courses.

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