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.
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.
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.
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.
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.
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.
> 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.
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.
> 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.
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 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.
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]
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.
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
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.
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.
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.
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.
> 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 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."
> 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.
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.
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]".
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).
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.
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.
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.
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.
>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:
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.
> Indeed, professional mathematicians are usually the ones teaching prospective high school teachers during undergrad, so we are the ones to solve it.
The vast majority of people who are paid to use their knowledge of mathematics aren't doing math research. In fact, mathematics research is so uneconomical that most mathematics researchers spend 20%+ of their job on teaching.
The number of people who are paid a full 12 (or even 9) month salary to do mathematics research is vanishingly small.
> This problem is accessible to anyone who knows basic high school algebra, but it requires a lot of effort
That effort is really difficult to get out of students because few students have intrinsic motivation, and there's very little extrinsic motivation (unless 3 post-docs is your idea of a good way to spend your 30s...)
I'm not sure what the solution is, but showing more people what mathematics researchers in universities do with their non-teaching time seems like the wrong solution.
What I described is what I consider to be "doing mathematics". As distinct from applying mathematics to other problems (also a very worthwhile endeavor!)
I'm certainly not encouraging everyone to pursue a career as a math professor; the job market is poor, and is getting bleaker.
But I feel like prospective math high school teachers should have an experience like the one I described. Much more important, in my opinion, than learning how to row reduce matrices, do integration by parts, or prove the intermediate value theorem.
>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.
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.
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), 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.
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 :-/
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.
> 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.
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...
> 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.
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.
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.
> 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.
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.
The key difference that's being overlooked between math teachers and football coaches are the populations they are working with. Football teams consist of competitive kids that WANT to play the sport and perform well (win). On the other hand most math classrooms consist of kids that HAVE to be there against their will, regardless of their interest in math.
I have to confess I haven't read the article. My first thought though was this would work only if the participants in Maths lessons are as keen as football players. Football(rugby in my case) is fun, even now as a middle-aged man I have vivid memories of playing rugby but hardly any memories cracking hard maths problems. I was motivated to play sport, it was fun. Maths was fun but not to the same level as a sport.
Casual phone games have more mathematics involved than people appreciate. Even if that involves figuring out that you will die before completing the game unless you pay for it.
But, regardless of their merits, casual games are insanely popular for various reasons.
And not just casual games. All sorts of games involve mathematics.
That may not be your case, but a lot of people have fond memories playing SimCity or min-maxing their paladin.
Thanks for your response. Certainly gave me a different perspective. More so because I am not a gamer. Not sure why but I never caught the gaming bug but most people I went to college with dabbled in games.
Former math teacher here and basketball coach. I agree with the article. My training as a teacher emphasized fostering investment from students just as much as it did lesson planning. All of these things are are really important.
But one of the toughest problems in teaching math is that, unlike sports, a student doesn't get a whole lot of feedback on their success and improvement in the short-term, besides grades. And grades have some major cons when it comes to motivation. Also, most of us don't use the exact skills from math class in life outside of class. What we use is the mindset of finding first principles, of abstraction, of modeling, and of problem solving. It's quite difficult to connect these skills to the mentality of kids and adolescents.
This is where extracurriculars come in, in my opinion. They teach the meta-skills of improvement mindset, coachability, and work ethic. I saw first-hand how my players came to class with a different mindset when sports were in season.
I have to say, it sounds like Mr. Urschel benefited from having good sports coaches, while having math teachers that maybe weren't so good at the motivational part. A lot of folks have the opposite mix. So I wouldn't say there's necessarily an underlying thread that generalizes to all of American education.
The workload of a football coach is probably much higher than that of a mathematics teacher. They are up and ready for morning practice early and stay late after school for more practice. They review videotapes of games, create strategy , and travel with students.
I doubt there are many mathematics teachers that have the same dedication for the success of their cohort as football occurs , on average.
Absolutely. The two roles are both challenging, but math teachers don't typically have kids competing to be in their class. It's a lot easier to motivate someone when they have someone on the bench who wants their spot.
Up early to make sure the classroom is ready. Stay late to help struggling students. Lesson planning. Multiple groups of students per day, each of which might be on a slightly different topic. Far fewer pep rallies.
source: Live with a math teacher. Child of two teachers, one of which taught for 18 years in a Brooklyn public middle school.
Exactly. That's also why the phrase" let's run government like a business" doesn't work. You can't just lay off 10% of your country's population if they don't perform.
In K-12 schools, most coaches are also teachers. I was a math teacher and basketball coach, for instance. When season was in, some of my days were 14+ hours long.
Does the team's performance affect the football coach's job evaluation/progression? And is this the case in every school in the country? I've gathered (from popular media) that high school football is taken very seriously in some parts of the country, but I mean outside of those areas.
Note: I'm aware college football coaches are fired if on-field results are below-par.
Math teams are a thing. They even have a coach just like football. Also football gets played in regular physical education also, with much less individual attention to students.
Yup, I was on the math team at my HS and we won state my senior year. Certainly not as rah-rah as football (which I was also in) but it was competitive and we all took it pretty seriously (obviously). Winning the state competition was just as satisfying than any playoff victory I had in football (perhaps more so).
You mean, they should only have to deal with people who are motivated enough to volunteer to spend extra, non-mandated time at school and who are supported enough that someone is paying extra money for them to participate, in an activity that while not mandatory produces broader social approbation for success than math?
Football coaches aren't generally better at motivating people than math teachers (heck, they are sometimes the exact same person.) They are just operating in a context that favors having more motivated people to start with and accepts them further weeding out the unmotivated.
I agree with the general idea that coaches are able to provide a much more personal and thorough training than the traditional teacher, but that's not a good thing for students and society in general. The reason is because students should have a right and an expectation to equal access to education. The reason we see very little diversity in the history of mathematics and science is because of this unequal access to mentors in particular, and education in general. (Most of the outstanding non-white and/or non-male mathematicians had white male mentors or sponsors that helped them get into the Old Boys Club.) When you have a monoculture in a field, then people tend to follow societal and cultural norms. This tendency impedes progress at best, and can cause disastrous blindspots at worst. The current American educational systems is an imperfect attempt at providing all students with equal opportunites. If we'd like to improve upon this, then let us improve the teacher to student ratio, rather than leaving it up to new hybrid teacher-coaches to try to identify MVPs at a young age, potentially allowing late-bloomers and marginalized groups to slip through the cracks.
I think that exceptional athletes should get more coaching than regular-old-me should get Phys Ed. I think that exceptional math students should get more coaching than regular-old student who needs to know basic math to survive in life. No different than music, art, science, or sports; I don't see a reason to put a thicker, even coat of peanut butter across all those disciplines and all the students, but rather to tailor the experience to go deeper in select areas when a student shows aptitude and interest.
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.
I'm a math professor, a white male, I was something of a child prodigy, and I was enormously blessed with math coaches who went far out of their way to help me.
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 really only disagree with your characterization of my point: "Because opportunities aren't equally available to everyone, they shouldn't be made available to anyone?"
I'm not saying opportunities shouldn't be made available to anyone, just that a stronger mentorship system than we already have in scientific higher-education is likely to reinforce the trend of white-male dominance that we already have, rather than temper it.
>The reason is because students should have a right and an expectation to equal access to education.
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.
Totally agree! I was in SMPY (700+ on “old SAT” math at under age 13) and did a self-study and examination to learn and test out of Algebra 1, Geometry, and Algebra 2 and start Algebra 3 during a 3 week summer camp/course.
It was one of the best academic experiences I had prior to college. No BS, just self-study (with mentor TAs present if needed) and test when done (and crucially, done at your pace, not that of the slowest target kid in the class).
Only downside was that meant I’d taken all the high school math by the end of 10th grade (5 on AP Calc AB [what we offered]) and so had to commute part-day to Univ of MD.
I know that’s an outlier case, but damn regular math class was 5-20x too slow to hold my interest. I don’t wish to inflict that on others in whatever their field of interest and abilities are.
Agreed. Math in particular is an area where mentorship matters a lot. Pretty tautologically, math relies on correctness, and correctness can be tricky to grok on your own. On the other hand, if you're trying to paint, make music, or write a story, there's a much larger margin for error and subjectivity, and it's easier to intuit when something feels wrong.
There's a reason we talk about amateur mathematician cranks and not amateur pianist cranks.
I think the fundamental difference here is that football is inherently competitive while math is not. While it's true that football coaches have incentives to produce great players that will eventually play in college/professionally, the most immediate incentive is to have players that will win games. Math teachers have no similar competitive incentives. The "passion" that Urschel mentions here is probably largely a product of the desire football coaches have to win every Friday/Saturday night, and this is reflected in the way that they develop kids. Additionally, Math teachers have to both teach kids AND grade them. This limits how much a teacher can really be "on your team" because they need to retain the ability to give you a bad grade. This is true with football coaches as well, but with football there is always a bigger "bad guy" (other teams/players) so a good coach is always more aligned with and close to the players.
“... it didn’t stop my coaches from encouraging me to believe I could reach my goal, and preparing and pushing me to work for it. When they told me I had potential but would have to work hard, I listened. I heard their voices in my ear when I dragged myself out of bed for predawn weightlifting sessions ...”
How many adults can clearly communicate the incentives for predawn calculus sessions?
"You'll have a chance at landing the dream job of the richest person on Earth"? (Bill Gates, member of the technical staff at Bell Labs in the late 1940s.)
I'm sure math teachers would love to get a million dollar salary. Not quite sure whether they would be willing to leave their students with irreversible brain damage for the sake of their career, though.
Eh. Sure, students can be disinterested. But, in my personal experience, in advanced high school courses (where the students were highly engaged) at a top-notch public school, the math teachers were, by and large, far less enthusiastic than the teachers of all the other subjects.
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.
That probably has a lot to do with it. It's hard to maintain enthusiasm over the years if it isn't being returned. I think the lack of passion from the students is more about the voluntary nature of the football team than anything inherent to the jocks.
The rewards are different, too. There is no level of mathematical achievement that will get the kind of broad (but typically local) social status and public approval of being a starting high school quarterback. Math just doesn't tap into anything near as easy and primal.
That's true, but I've seen years where an enthusiastic professor gets a packed room for a normally undersubscribed elective course like computational complexity theory or advanced algorithms.
True, but there's a heavy selection going on there. A packed complexity class might mean 50? enthusiastic undergrads out of several hundred math/cs students at a university.
A dedicated and charismatic professor might be instrumental in turning an interested computer science student into a raging algorithmist (indeed, I actually think a lot of computer science grad students study what they study because somebody appropriately passionate introduced it to them), but converting a just-here-for-the-requirement high schooler is a different story.
The football team also has a heavy selection bias.
Personally, I got into CS because there was a required intro class at my undergrad. A dedicated and charismatic professor noticed my aptitude and encouraged me to take the next course and consider a CS major. This could easily have happened in high school if my high school had a required CS course.
Perform brain imaging, split students into those with a strong math co-processor brain and those without. For those without, teach them math using logic rather than rote learning which is obvious to those with a strong math brain.
What an absurd, elitist, reductive, sci-fi response. Brains don't have ALUs in them, btw. Nor can we image for different skills. Nor should we subject people to this kind of dystopian hierarchy of faulty biological determinism.
I agree, perhaps brain scans aren't an option, but I do think two tracks based on some kind of testing makes a lot of sense.
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.
You don't need brain imaging to sort out those with a natural talent and affinity for math, the challenge of an education system is to maximize the level of those without such affinities.
That's not how football is approached. I think GP's point is that football is approached by selecting for those with the most potential and maximizing that and that math teachers can't be like football coaches really.
- 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.