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>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 inher

by jasode 7y ago

>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 https://en.wikipedia.org/wiki/Dyscalculia#Common_symptoms [2] https://en.wikipedia.org/wiki/Terence_Tao https://en.wikipedia.org/wiki/Terence_Tao

jacobolus 7y ago
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).
- jasode 7y ago
  >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.
  - jacobolus 7y ago
    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.
    - jasode 7y ago
      >, 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.
TallGuyShort 7y ago
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.
- andrewem 7y ago
  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.)
  - TallGuyShort 7y ago
    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.
- jasode 7y ago
  >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.
littlestymaar 7y ago
> 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)…
- falcor84 7y ago
  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.
  - solveit 7y ago
    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.
    - littlestymaar 7y ago
      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.

by jasode 7y ago

jacobolus 7y ago

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).

jasode 7y ago
>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.
- jacobolus 7y ago
  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.
  - jasode 7y ago
    >, 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.

TallGuyShort 7y ago

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.

andrewem 7y ago
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.)
- TallGuyShort 7y ago
  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.
jasode 7y ago
>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.

littlestymaar 7y ago

> 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)…

falcor84 7y ago
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.
- solveit 7y ago
  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.
  - littlestymaar 7y ago
    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.