5 ms·

> 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 signi

> 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.

varjag 8y ago
And the effect is they improve 10 times faster than others who don't do all that.
- jacobolus 8y ago
  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.
  - pdpi 8y ago
    > 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.
- smokeyj 8y ago
  But it’s not “times faster”, it’s a linear function of time..
gameswithgo 8y ago
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?
- jacobolus 8y ago
  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.
  - pedrosorio 8y ago
    > nobody improves 10 times faster > At the least, nearly everyone is capable of successfully handling the standard school curriculum, given a few hours per week of 1:1 tutoring by an expert. Your second statement does not imply the first one. "Nearly everyone" may be capable of successfully handling the standard curriculum, and still, given the same resources, and hours/week some people may progress 10x faster than others.
    - jacobolus 8y ago
      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.
jasode 8y 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 8y 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 8y 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.
chongli 8y ago
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.
- jacobolus 8y ago
  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.
  - 8y ago
turndown 8y ago
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.
barry-cotter 8y ago
This is as absurd as saying that neurotypical children have more preparation for social interaction or better preparation than autistic children, or that Kalenjin children have more preparation for running than Khoisan ones. People vary enormously in their natural talents socially, academically and athletically. https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss > Gauss was a child prodigy. In his memorial on Gauss, Wolfgang Sartorius von Waltershausen says that when Gauss was barely three years old he corrected a math error his father made; and that when he was seven, he confidently solved an arithmetic series problem faster than anyone else in his class of 100 students.[9] Many versions of this story have been retold since that time with various details regarding what the series was – the most frequent being the classical problem of adding all the integers from 1 to 100. https://en.wikipedia.org/wiki/Terence_Tao https://en.wikipedia.org/wiki/Terence_Tao Tao exhibited extraordinary mathematical abilities from an early age, attending university-level mathematics courses at the age of 9. He and Lenhard Ng are the only two children in the history of the Johns Hopkins' Study of Exceptional Talent program to have achieved a score of 700 or greater on the SAT math section while just nine years old; Tao scored a 760.[9] Tao was the youngest participant to date in the International Mathematical Olympiad, first competing at the age of ten; in 1986, 1987, and 1988, he won a bronze, silver, and gold medal. He remains the youngest winner of each of the three medals in the Olympiad's history, winning the gold medal shortly after his thirteenth birthday.
- deleted 8y ago
  [deleted]

varjag 8y ago

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

jacobolus 8y ago
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.
- pdpi 8y ago
  > 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.
smokeyj 8y ago
But it’s not “times faster”, it’s a linear function of time..

gameswithgo 8y ago

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?

jacobolus 8y ago
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.
- pedrosorio 8y ago
  > nobody improves 10 times faster > At the least, nearly everyone is capable of successfully handling the standard school curriculum, given a few hours per week of 1:1 tutoring by an expert. Your second statement does not imply the first one. "Nearly everyone" may be capable of successfully handling the standard curriculum, and still, given the same resources, and hours/week some people may progress 10x faster than others.
  - jacobolus 8y ago
    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.

jasode 8y 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 8y 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 8y 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.

chongli 8y ago

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.

jacobolus 8y ago
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.
- 8y ago

turndown 8y ago

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.

barry-cotter 8y ago

This is as absurd as saying that neurotypical children have more preparation for social interaction or better preparation than autistic children, or that Kalenjin children have more preparation for running than Khoisan ones. People vary enormously in their natural talents socially, academically and athletically. https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss > Gauss was a child prodigy. In his memorial on Gauss, Wolfgang Sartorius von Waltershausen says that when Gauss was barely three years old he corrected a math error his father made; and that when he was seven, he confidently solved an arithmetic series problem faster than anyone else in his class of 100 students.[9] Many versions of this story have been retold since that time with various details regarding what the series was – the most frequent being the classical problem of adding all the integers from 1 to 100. https://en.wikipedia.org/wiki/Terence_Tao https://en.wikipedia.org/wiki/Terence_Tao Tao exhibited extraordinary mathematical abilities from an early age, attending university-level mathematics courses at the age of 9. He and Lenhard Ng are the only two children in the history of the Johns Hopkins' Study of Exceptional Talent program to have achieved a score of 700 or greater on the SAT math section while just nine years old; Tao scored a 760.[9] Tao was the youngest participant to date in the International Mathematical Olympiad, first competing at the age of ten; in 1986, 1987, and 1988, he won a bronze, silver, and gold medal. He remains the youngest winner of each of the three medals in the Olympiad's history, winning the gold medal shortly after his thirteenth birthday.

deleted 8y ago
[deleted]