I don’t remember a period of my life where I didn’t feel like I had a deep understanding of math, and so it’s hard for me to separate out mathematical ability and cognitive ability.
I’ve also seen advice from a handful of places I respect to learn as much math as you can stand, because there often is transfer from mathematical topics to practical applications. This is much more true for engineers, physicists, and software developers than it is for people in other professions, but still suggests that the first negative consideration you raise is strong (unless it doesn’t apply to you).
Reduced need for memorization (while learning math). When you understand math deeply, you see how many different mathematical problems are special cases of a single more general problem, so that in order to remember how to do all of the problems, it suffices to remember the solution to that more general problem.
I remember talking with a friend in high school about physics of electromagnetism. He had the poor fortune to take the non-calculus based version of physics, and so he had to memorize various geometries and the electric potentials they created. I was horrified- in calculus-based physics, we learned one law and then integrated as necessary.
I don’t remember a period of my life where I didn’t feel like I had a deep understanding of math, and so it’s hard for me to separate out mathematical ability and cognitive ability.
I’d be interested in hearing more about your experience. A lot of smart people don’t develop a deep understanding of math because that’s not how the subject is taught and because they don’t have the initiative to try to work things out themselves. With this in mind, to what do you attribute your success?
Hope this isn’t too off-topic, but I wonder if you have any ideas about why that is.
The main impediment to many far-mode thinkers learning hard (post-calculus) math is the drill and drudgery involved. If you’re going to learn hard math, it seems you should, by all means, learn it deeply. That’s not the obstacle. The obstacle is that to learn math deeply, you must first learn a lot of it rotely—at least the way it’s taught.
In the far-distant past, when I was in school, learning elementary calculus meant rote drilling on techniques of solving integrals. Is this still the case? Is it inevitable, or is it the result of methods of education?
The main reason “smart people” avoid math isn’t that they want to avoid depth; rather, what is, at least for some of them, drudgery. Math, more than any subject I know of, seems to require a very high level of sheer diligence to get to the point where you can start thinking about it deeply. Is this inevitable?
Hope this isn’t too off-topic, but I wonder if you have any ideas about why that is.
I think that the point is that more people are capable of routine tasks than of conceptual understanding, and that educational institutions want lots of people to do well in math class on account of a desire for (the appearance of) egalitarianism.
In the far-distant past, when I was in school, learning elementary calculus meant rote drilling on techniques of solving integrals. Is this still the case?
What time period was this? (No need to answer if you’d prefer not to :-) )
Math, more than any subject I know of, seems to require a very high level of sheer diligence to get to the point where you can start thinking about it deeply. Is this inevitable?
Some diligence is necessary, but not as much as it appears based on standard pedagogy. I wish that I could substantiate this in a few lines. If you say something about what math you know/remember, I might be able to point you to some helpful references.
In the far-distant past, when I was in school, learning elementary calculus meant rote drilling on techniques of solving integrals. Is this still the case? Is it inevitable, or is it the result of methods of education?
Some degree of this is probably inevitable. Integration in particular has no closed solution (unlike differentiation), so there really is no one general method you can apply to all problems. All you can do is remember a bag of tricks. While for differentiation, a few general rules allow you to integrate all elementary and trigonometric functions, and that’s pretty much all you encounter in school.
With this in mind, to what do you attribute your success?
Well, looking back I have to attribute a lot of my perception of success with blindness, in the sense that 5-6 year old me thought he was a hot math talent because he knew about integers when the teacher was teaching the class about natural numbers. (I still remember raging against the claim that the right answer was “you can’t subtract 3 from 2!” instead of “negative 1!”) From what I can tell from looking at curriculum online, that’s ~5 years ahead of schedule but I’d interpret that as the curriculum putting it late (though, on reflection, that could be Dunning-Kruger).
I remember jumping ahead of (well, deeper than- below?) the curriculum frequently, and suspect that it had different causes in different circumstances. Rapid calculation is probably just high g, but rapid perception of concepts and connections probably has something to do with intuition or vision that I find difficult to articulate.
I’ve also never been particularly good at explaining why I know what I know with regards to math- from refusing the step through the algebra when I could solve a problem in my head, to avoiding college classes which were primarily about proving that methods worked (i.e. calculus the second time around) rather than introducing new methods. I have, through deliberate practice, gotten better at writing proofs in the last year or two, but still regularly come across simple theorems where I say “I know X is true, but don’t know how to show X is true.”
I do think I would have been more successful in a Moore method environment which is designed to teach a deep understanding of mathematics- it seems likely to me_now that me_past would have learned/wanted to care about rigor much earlier in that sort of environment, and would have kept pushing my math boundaries much more uniformly.
I don’t remember a period of my life where I didn’t feel like I had a deep understanding of math, and so it’s hard for me to separate out mathematical ability and cognitive ability.
I’ve also seen advice from a handful of places I respect to learn as much math as you can stand, because there often is transfer from mathematical topics to practical applications. This is much more true for engineers, physicists, and software developers than it is for people in other professions, but still suggests that the first negative consideration you raise is strong (unless it doesn’t apply to you).
I remember talking with a friend in high school about physics of electromagnetism. He had the poor fortune to take the non-calculus based version of physics, and so he had to memorize various geometries and the electric potentials they created. I was horrified- in calculus-based physics, we learned one law and then integrated as necessary.
I’d be interested in hearing more about your experience. A lot of smart people don’t develop a deep understanding of math because that’s not how the subject is taught and because they don’t have the initiative to try to work things out themselves. With this in mind, to what do you attribute your success?
Hope this isn’t too off-topic, but I wonder if you have any ideas about why that is.
The main impediment to many far-mode thinkers learning hard (post-calculus) math is the drill and drudgery involved. If you’re going to learn hard math, it seems you should, by all means, learn it deeply. That’s not the obstacle. The obstacle is that to learn math deeply, you must first learn a lot of it rotely—at least the way it’s taught.
In the far-distant past, when I was in school, learning elementary calculus meant rote drilling on techniques of solving integrals. Is this still the case? Is it inevitable, or is it the result of methods of education?
The main reason “smart people” avoid math isn’t that they want to avoid depth; rather, what is, at least for some of them, drudgery. Math, more than any subject I know of, seems to require a very high level of sheer diligence to get to the point where you can start thinking about it deeply. Is this inevitable?
I think that the point is that more people are capable of routine tasks than of conceptual understanding, and that educational institutions want lots of people to do well in math class on account of a desire for (the appearance of) egalitarianism.
What time period was this? (No need to answer if you’d prefer not to :-) )
Some diligence is necessary, but not as much as it appears based on standard pedagogy. I wish that I could substantiate this in a few lines. If you say something about what math you know/remember, I might be able to point you to some helpful references.
Some degree of this is probably inevitable. Integration in particular has no closed solution (unlike differentiation), so there really is no one general method you can apply to all problems. All you can do is remember a bag of tricks. While for differentiation, a few general rules allow you to integrate all elementary and trigonometric functions, and that’s pretty much all you encounter in school.
Well, looking back I have to attribute a lot of my perception of success with blindness, in the sense that 5-6 year old me thought he was a hot math talent because he knew about integers when the teacher was teaching the class about natural numbers. (I still remember raging against the claim that the right answer was “you can’t subtract 3 from 2!” instead of “negative 1!”) From what I can tell from looking at curriculum online, that’s ~5 years ahead of schedule but I’d interpret that as the curriculum putting it late (though, on reflection, that could be Dunning-Kruger).
I remember jumping ahead of (well, deeper than- below?) the curriculum frequently, and suspect that it had different causes in different circumstances. Rapid calculation is probably just high g, but rapid perception of concepts and connections probably has something to do with intuition or vision that I find difficult to articulate.
I’ve also never been particularly good at explaining why I know what I know with regards to math- from refusing the step through the algebra when I could solve a problem in my head, to avoiding college classes which were primarily about proving that methods worked (i.e. calculus the second time around) rather than introducing new methods. I have, through deliberate practice, gotten better at writing proofs in the last year or two, but still regularly come across simple theorems where I say “I know X is true, but don’t know how to show X is true.”
I do think I would have been more successful in a Moore method environment which is designed to teach a deep understanding of mathematics- it seems likely to me_now that me_past would have learned/wanted to care about rigor much earlier in that sort of environment, and would have kept pushing my math boundaries much more uniformly.