4 ms·

How is that relevant? Many of us are not going to spend our entire lives in a university.

by foo101 9y ago

jacobolus 9y ago
The general way to master a large pile of difficult abstractions is to study for years, ideally in a context focused on that study. This can be done independently of any expert feedback (e.g. by reading books, watching lecture videos, and doing a big pile of ungraded self-assigned homework problems), but it’s not easy or typically very efficient. For the majority of people who want to deeply learn mathematical topics but who are starting from a low level, the most efficient and effective advice for them is to go attend a university. An online discussion forum isn’t typically going to cut it, IMO. People won’t be at the same level, won’t be working on the same topics, won’t have the same goals and interests, the same level of commitment, coordinated schedules, etc. This has been tried many times, but I have not heard many successful reports. Online discussion forums are great for handling independent limited-scope questions–answers (including for current university students), and okay for coordinating research efforts among people who are already experts, but in general are not very effective for getting a group of people to diligently work through years of prerequisite material in an organized fashion. You shouldn’t spend your life in a university, but if you have a goal of learning advanced mathematics, you might benefit from spending a few years there. YMMV. As an alternative, if you can find an expert tutor/coach to meet with one-on-one on a regular basis, that is even better than a formal course. But that is expensive and hard to scale.
- jimhefferon 9y ago
  Very well put. > it’s not easy or typically very efficient It is also very possible for folks to think they have it right, and go off in entirely wrong directions. In my experience it is likely, even in quite good and hard-working students. I teach an inquiry-based proofs class and so I sit through the students's discussion. Even on material that would be considered trivial in many of the books mentioned here, there are people saying aloud things in that discussion that are just simply wrong. It is not that these folks are dummies, it is that this stuff is hard. I'm not a people person but one thing I've seen from this class is that a lot of learning is about community. It makes a big difference.
- mindcrime 9y ago
  You shouldn’t spend your life in a university, but if you have a goal of learning advanced mathematics, you might benefit from spending a few years there. YMMV. The problem is, that simply isn't possible / practical for many people. Assuming we're talking about people who are career professionals who are already past their first (and maybe only) stint at a university, most people just don't have the free time (and other resources) to sit in classes at a university. Especially since most universities still cater almost exclusively to "traditional" students who take classes in person, during the day. If universities made more night classes available, or more online/hybrid classes, I could buy this. But for a lot of us, the best we can do is some combination of "watch Youtube videos, do MOOC's, self-study, do ungraded problems, and ask for help on Internet forums". That said, not all problems you do on your own have to be exactly "ungraded". For many mathematical subjects you can find books of exercises that include the answers so you can "grade" yourself (assuming you can resist the temptation to cheat and look at the answers first), or you can find previous years problem sets and exams on many course websites. Yeah, learning the stuff outside of a university setting is probably harder in some sense, but it's not impossible. And if that avenue is the only one available to somebody, then it is - by definition - the best avenue available to them.
  - deleted 9y ago
    [deleted]
  - jacobolus 9y ago
    Most of those people are not going to get all that far learning advanced mathematics, unless they spend serious amounts of time and effort. But sure, I wouldn’t want to discourage them from giving it a shot. Personally I am not an academic, I never went to grad school, and I do lots of self-studying of various technical topics, up to hopefully some reasonable level of understanding.
dsacco 9y ago
This might sound callous, but most of us are not going to master advanced mathematics or theoretical computer science. If you want to self-study, all the material you could ever hope to need is available in a mulitude of books targeted to various levels of prior experience and with many different styles of exposition. If you want to study with others, this is precisely what college is for. People who don’t want to study in a college setting basically don’t want to study in a group setting in general, because the college setting actually is the most efficient group setting. I think most people underestimate the sheer vastness of the material you have to learn and the requisite time you need to learn it. Reading isn’t enough, you have to do it - over and over. It’s frankly a slog, and there’s no more efficient way than being immersed in coursework. The entire benefit of a course structure is the instructor. The community forum helps, but really it’s an extension of the instructor. A qualified instuctor can quickly resolve a complex question that might take you hours or days to (maybe incorrectly) answer yourself. If you’re capable of learning genuinely unfamiliar math without an instructor, that means you have sufficient mathematical maturity that you don’t need a group forum. At that point it’s actually more efficient to self-study. I get the sense people believe mathematicians are pretentious when they say things like this, but I think that’s uncharitable. They typically give advice like this because they have a significant amount of personal experience and know very well what tends to work (and how efficiently it works).

How is that relevant? Many of us are not going to spend our entire lives in a university.

by foo101 9y ago

How is that relevant? Many of us are not going to spend our entire lives in a university.

The general way to master a large pile of difficult abstractions is to study for years, ideally in a context focused on that study. This can be done independently of any expert feedback (e.g. by reading books, watching lecture videos, and doing a big pile of ungraded self-assigned homework problems), but it’s not easy or typically very efficient. For the majority of people who want to deeply learn mathematical topics but who are starting from a low level, the most efficient and effective advice for them is to go attend a university. An online discussion forum isn’t typically going to cut it, IMO. People won’t be at the same level, won’t be working on the same topics, won’t have the same goals and interests, the same level of commitment, coordinated schedules, etc. This has been tried many times, but I have not heard many successful reports. Online discussion forums are great for handling independent limited-scope questions–answers (including for current university students), and okay for coordinating research efforts among people who are already experts, but in general are not very effective for getting a group of people to diligently work through years of prerequisite material in an organized fashion. You shouldn’t spend your life in a university, but if you have a goal of learning advanced mathematics, you might benefit from spending a few years there. YMMV. As an alternative, if you can find an expert tutor/coach to meet with one-on-one on a regular basis, that is even better than a formal course. But that is expensive and hard to scale.

jimhefferon 9y ago
Very well put. > it’s not easy or typically very efficient It is also very possible for folks to think they have it right, and go off in entirely wrong directions. In my experience it is likely, even in quite good and hard-working students. I teach an inquiry-based proofs class and so I sit through the students's discussion. Even on material that would be considered trivial in many of the books mentioned here, there are people saying aloud things in that discussion that are just simply wrong. It is not that these folks are dummies, it is that this stuff is hard. I'm not a people person but one thing I've seen from this class is that a lot of learning is about community. It makes a big difference.
mindcrime 9y ago
You shouldn’t spend your life in a university, but if you have a goal of learning advanced mathematics, you might benefit from spending a few years there. YMMV. The problem is, that simply isn't possible / practical for many people. Assuming we're talking about people who are career professionals who are already past their first (and maybe only) stint at a university, most people just don't have the free time (and other resources) to sit in classes at a university. Especially since most universities still cater almost exclusively to "traditional" students who take classes in person, during the day. If universities made more night classes available, or more online/hybrid classes, I could buy this. But for a lot of us, the best we can do is some combination of "watch Youtube videos, do MOOC's, self-study, do ungraded problems, and ask for help on Internet forums". That said, not all problems you do on your own have to be exactly "ungraded". For many mathematical subjects you can find books of exercises that include the answers so you can "grade" yourself (assuming you can resist the temptation to cheat and look at the answers first), or you can find previous years problem sets and exams on many course websites. Yeah, learning the stuff outside of a university setting is probably harder in some sense, but it's not impossible. And if that avenue is the only one available to somebody, then it is - by definition - the best avenue available to them.
- deleted 9y ago
  [deleted]
- jacobolus 9y ago
  Most of those people are not going to get all that far learning advanced mathematics, unless they spend serious amounts of time and effort. But sure, I wouldn’t want to discourage them from giving it a shot. Personally I am not an academic, I never went to grad school, and I do lots of self-studying of various technical topics, up to hopefully some reasonable level of understanding.

dsacco 9y ago

This might sound callous, but most of us are not going to master advanced mathematics or theoretical computer science. If you want to self-study, all the material you could ever hope to need is available in a mulitude of books targeted to various levels of prior experience and with many different styles of exposition. If you want to study with others, this is precisely what college is for. People who don’t want to study in a college setting basically don’t want to study in a group setting in general, because the college setting actually is the most efficient group setting. I think most people underestimate the sheer vastness of the material you have to learn and the requisite time you need to learn it. Reading isn’t enough, you have to do it - over and over. It’s frankly a slog, and there’s no more efficient way than being immersed in coursework. The entire benefit of a course structure is the instructor. The community forum helps, but really it’s an extension of the instructor. A qualified instuctor can quickly resolve a complex question that might take you hours or days to (maybe incorrectly) answer yourself. If you’re capable of learning genuinely unfamiliar math without an instructor, that means you have sufficient mathematical maturity that you don’t need a group forum. At that point it’s actually more efficient to self-study. I get the sense people believe mathematicians are pretentious when they say things like this, but I think that’s uncharitable. They typically give advice like this because they have a significant amount of personal experience and know very well what tends to work (and how efficiently it works).