I'm not necessarily convinced of Murray's example of addition vs calculus here.
He could be correct, but couldn't it be possible that being able to get above a zero on a calculus test is really the same exact same thing as scoring in the top X percentile on a test of addition?
Imagine we had an apple picking competition. We give a large number of people their own apple tree from which to pick apples. If the lowest apples on these trees are found 4 feet above the ground, then almost everyone would be able to pick at least some apples with only those at the extreme low end of height being unable to pick any apples. Thus, plotting a histogram of people collecting apples would likely very nearly have a bell curve.
Now, what if the same test was performed except this time the lowest apples on the tree are at 8 feet above the ground. This would eliminate a large number of people under roughly 6 feet tall who cannot reach 8 feet, and it would appear from this test that there is some special ability of these people who picked >0 apples compared to those who couldn't pick apples.
However, really both of these tests are more or less measuring a single variable: height. Height is a continuous variable, but at first glance appears that there might be a discrete variable in the second apple picking competition.
Could this not also be the case with Murray's addition and calculus tests with the general cognitive factor taking the place of height?
I think the problem with your example (wonderful as it is!) is that it's not capturing the essence of what's happening. I also think talking about discrete versus continuous variables doesn't add much, but I may well be wrong about that.
First, I think that getting above zero on a calculus test is tapping into the same underlying construct (in this case, g) as getting a certain mark on an arithmetic test. However, the reason calculus is a difference in kind is that it relies on a different set of abilities and mental modelling that is biologically barred to the bottom 20%, perhaps even the bottom third. I think it's appropriate to call this level of symbolic thought a difference in kind.
Perhaps if we want to be pedantic we can say that all intellectual differences are ones of degree because it's mostly difference in g which determines capability. But I think this isn't useful and doesn't help us to understand biological limitations.
Also, if most children in the bottom 20% cannot ever learn to answer "differentiate y = 4x - 2 sin x", I think it's useful for teachers, parents, and school officials to recognise this as a difference in kind, because they tend to harbour the illusion that if little Freddy did okay in multiplication, he should be able to learn a bit of everything.
The depressing part of your argument is well known to me, but I still find myself fascinated by people like von Neumann, or Feynman, or any of the great ones, really. I studied physics at Purdue, and while I did well, I knew by the end of four years that I was never going to be able to go on to actually do physics. I met people who were so much brighter than the rest of us. For example, in my advanced mechanics class there were ten of us. The final grades in our class were 99, 69, 49, 48...14. (I was the 48.) The guy with the 99 only came to class to take the (fiendishly difficult) tests. We wondered if he was even human. I knew that I was getting a glimpse of what was possible, but also that I would never be able to go there - and I have an IQ of 141. I would probably need another 15-20 points to do the kinds of things I once dreamed of. The world is a better place for the work of a very few highly gifted people - we need to find them when they're young and help them reach their potential.
I'm not necessarily convinced of Murray's example of addition vs calculus here.
He could be correct, but couldn't it be possible that being able to get above a zero on a calculus test is really the same exact same thing as scoring in the top X percentile on a test of addition?
Imagine we had an apple picking competition. We give a large number of people their own apple tree from which to pick apples. If the lowest apples on these trees are found 4 feet above the ground, then almost everyone would be able to pick at least some apples with only those at the extreme low end of height being unable to pick any apples. Thus, plotting a histogram of people collecting apples would likely very nearly have a bell curve.
Now, what if the same test was performed except this time the lowest apples on the tree are at 8 feet above the ground. This would eliminate a large number of people under roughly 6 feet tall who cannot reach 8 feet, and it would appear from this test that there is some special ability of these people who picked >0 apples compared to those who couldn't pick apples.
However, really both of these tests are more or less measuring a single variable: height. Height is a continuous variable, but at first glance appears that there might be a discrete variable in the second apple picking competition.
Could this not also be the case with Murray's addition and calculus tests with the general cognitive factor taking the place of height?
Thanks for the thoughtful comment, Hunter.
I think the problem with your example (wonderful as it is!) is that it's not capturing the essence of what's happening. I also think talking about discrete versus continuous variables doesn't add much, but I may well be wrong about that.
First, I think that getting above zero on a calculus test is tapping into the same underlying construct (in this case, g) as getting a certain mark on an arithmetic test. However, the reason calculus is a difference in kind is that it relies on a different set of abilities and mental modelling that is biologically barred to the bottom 20%, perhaps even the bottom third. I think it's appropriate to call this level of symbolic thought a difference in kind.
Perhaps if we want to be pedantic we can say that all intellectual differences are ones of degree because it's mostly difference in g which determines capability. But I think this isn't useful and doesn't help us to understand biological limitations.
Also, if most children in the bottom 20% cannot ever learn to answer "differentiate y = 4x - 2 sin x", I think it's useful for teachers, parents, and school officials to recognise this as a difference in kind, because they tend to harbour the illusion that if little Freddy did okay in multiplication, he should be able to learn a bit of everything.
The depressing part of your argument is well known to me, but I still find myself fascinated by people like von Neumann, or Feynman, or any of the great ones, really. I studied physics at Purdue, and while I did well, I knew by the end of four years that I was never going to be able to go on to actually do physics. I met people who were so much brighter than the rest of us. For example, in my advanced mechanics class there were ten of us. The final grades in our class were 99, 69, 49, 48...14. (I was the 48.) The guy with the 99 only came to class to take the (fiendishly difficult) tests. We wondered if he was even human. I knew that I was getting a glimpse of what was possible, but also that I would never be able to go there - and I have an IQ of 141. I would probably need another 15-20 points to do the kinds of things I once dreamed of. The world is a better place for the work of a very few highly gifted people - we need to find them when they're young and help them reach their potential.
Well said. Such profound relevance to our Western malaise...and yet so little reaction.
Someone who believes it is a difference in KIND should support Montessori as an educational method.
Someone who believes it is a difference in DEGREE should also support Montessori as an educational method.
Yo. How did you add more tabs to your substack home page than the 'home', 'archive', & 'about' tabs?