>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 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.