7 ms·

Here's my personal opinion about how you should approach this. It's all well and good to want to cover undergraduate math courses. When you are actually enroll

by abnry 8y ago

Here's my personal opinion about how you should approach this. It's all well and good to want to cover undergraduate math courses. When you are actually enrolled in a university, you will have enough inertia and motivation to complete the courses. However, when you are self-studying you are doing it all on your own. It's hard to be as thorough and cover everything. And so I ask, what really is your goal here? You don't have to learn everything about mathematics, because that is in fact impossible. My advice is to FIRST construct a bunch of projects, tasks or goals that require knowledge. It could be something like (a) implement a machine learning algorithm to do X from scratch (b) implement a simple physics engine (c) try to verify a number theory conjecture (d) be able to solve all the exercises in a book (e) be able to write up a compelling description/theorem/problem in math (d) numerically solve the quantum mechanics equations of a certain system Spend some time on material that will inspire you first to help get these goals. Numberphile on YouTube, or any of Brady Haran's videos, is a good place to start. But make the goals your own and make them personal. Math is not a spectator's sport. Make sure to DO mathematics, not just LEARN mathematics.

foo101 8y ago
Is this sentence incomplete or am I missing the context of the word "compelling" here? > be able to write up a compelling Can someone clarify what this means?
- mindcrime 8y ago
  I suspect it was meant to say "be able to write up a compelling proof".
- abnry 8y ago
  By mistake I swapped lines in my post when composing it. It is fixed now.
joe_the_user 8y ago
The OP did specify what he wanted - the basic undergraduate and starting graduate curriculum. That's a pretty well defined area: Algebra, Real Analysis, Geometry and Topology with maybe some complex analysis, number theory, statistics, CS or etc thrown in. I personally did work myself up to the graduate in math during the last two years of High School & first year of college. I was motivated by exploring ideas and gaining knowledge. I would guess that each person has a somewhat unique motivation strategy. Maybe solving problems gets some people doing stuff. I'm sure simply learning stuff can motivate others. Probably each person has to experiment to discover what works for them - I would pick up a calculus book and read it - well, I'd skim repeatedly and then read in depth, solving a few problems. Math is difficult, of course, so having a bit of patience with your until it gets the ideas on it's own is probably necessary.
- throwawaymath 8y ago
  > I personally did work myself up to the graduate in math during the last two years of High School & first year of college. ...how in the world did you manage to do this? Did you actually self-study, or were you placed in a gifted program? Self-studying all of undergraduate mathematics is more impressive than actually studying all of it in a four year classroom setting. Doing so as a teenager is amazing. The most gifted person I've ever known in mathematics is actually a physicist who entered Harvard at 16. He was already taking undergraduate math courses at 13/14 and had mastered all the undergraduate material by the first semester of his undergraduate degree. For the remaining years of his undergraduate degree he took graduate math courses. But in order to do that he needed to not only be gifted; he was placed in a program for extremely gifted kids sponsored by Stanford University from his preteen years. I don't think your anecdote is a great comparison for the OP's expectations (or for calibrating advice they'd benefit from).
  - joe_the_user 8y ago
    How in the world did you manage to do this? Did you actually self-study, or were you placed in a gifted program? I read quite a bit on my own, I took some courses at UCLA through a high school scholars program (including the undergraduate honors seminar). The entirety of the undergraduate program might be a slight exaggeration but I was ready for graduate level courses when I got to Berkeley. I think going through the material requires determination, not necessarily being extremely gifted. But then, it seems like people at someone's gone through a bunch material say by that fact they're gifted. Thus having done this, one is tautologically gifted. I've never tested at the extremely gifted level but I'm doubtful of single-measures of intelligence regardless. Edit: I don't think your anecdote is a great comparison for the OP's expectations (or for calibrating advice they'd benefit from). Neither of us know the OP. It's kind of up to them to calibrate what process works for them. Scanning a lot of math until I found good, clear explanations worked well for me.
- analog31 8y ago
  >>> I would guess that each person has a somewhat unique motivation strategy. Indeed, speaking only for myself, I don't think that I had the self discipline to carry through with such a plan as a teenager. I needed the classroom environment to learn math & physics, which became my college majors. At the same time, I was able to teach myself programming and electronics quite easily for some mysterious reason. And I ended up combining all of those things in grad school. Also, I only had access to the books and resources of a typical suburb with a decidedly anti-intellectual culture. The officials at my high school refused to offer a calculus course because they said it would be "elitist."
forkandwait 8y ago
Personally, I think this is bad advice, because without an undergrad+ background, the projects above will either be impossibly frustrating or you will make up some crackpot bullshit. Plus, the undergraduate curriculum is its own reward.
- enraged_camel 8y ago
  Nah, a good way to learn new skills is to pick a destination and then figure out what steps you need to take to get there. This type of “top-down” learning can help one stay motivated through the most frustrating road blocks. This is especially important for self-learning, because unlike an undergrad setting, the person is on their own and can’t rely on peers.
  - forkandwait 8y ago
    Well, we disagree. Plus, taking on some crazy problem without background saps motivation as well.
    - ny8jjang 8y ago
      I think setting attainable goals and working towards it helps. It is very easy to lose motivation if you don't know what you are going for..
      - forkandwait 8y ago
        Depending on your background, you probably don't have enough information to pick a long-term goal anyway. I am afraid that sounds curmudgeonly, but I have also seen students shoot themselves in the foot because they decided they didn't need a class for their not very well informed goals.
        enraged_camel 8y ago
        >>Depending on your background, you probably don't have enough information to pick a long-term goal anyway. Nah, it's totally possible for newbies to pick high-level long-term goals. This can be something like "I want to teach my computer to tell apart dogs and cats", or "I want to create a website where people can buy and sell yarn." From there, Google searches can direct someone towards concepts and various methods of learning them. I mean, you can disagree all you want, but this is in fact how many people learn things.
madhadron 8y ago
This is really good advice, and I thinking about suggesting the same thing. I've accepted that there are some basics that you have to have in place before you can really get traction and learn what you need to learn for specific projects.
40acres 8y ago
I think this is good advice. I tried to read parts of CLRS a few years ago but could not wrap my head around the work. The mathematical prerequisites needed for CLRS gave me an end point to guide my self learning to focus on specific areas: algebra (linear and elementary), calculus 1, combinatorics / probability, discrete math, propositional logic.
te_platt 8y ago
My oldest brother had a real love of mathematics and got a masters degree but never even applied to a PhD program. When I asked him about it he just said he was tired of working on other people's problems. He had a quiet programming position in a large company that gave him time to work on personal projects. Among other things he volunteered at the local schools teaching "fun" math. I guess I'm just agreeing with you about finding something that inspires you and go with it.
- selimthegrim 8y ago
  I don’t know what PhD program your brother was looking at but ideally candidacy committees won’t pass you unless you chose the problem or made it your own. Where your brother is right is that forming such a ideal committee is mostly politics.
  - Eridrus 8y ago
    I don't know about math, but in CS, a lot of funding comes from specific grants, so you'll be working on something related to what your advisor got grant funding for.
    - QuercusMax 8y ago
      Regardless of grants, etc., your general area of work is going to be based on what the faculty members' focuses are. But in any case, you'll probably still be doing your own sub-piece of some larger effort. You wouldn't choose an advisor at random, so this isn't really relevant. I mean, if you're interested in astrophysics, you shouldn't go to a school where the Physics department is focused on nuclear physics!
  - te_platt 8y ago
    I think you are right on both points. There were more details and his answer was just a quick way to say: He didn't care about the status of having a PhD He didn't need a PhD to advance his career He was having fun doing math stuff anyway
- 8y ago
MattHeard 8y ago
This is exactly the kind of advice I give to people who ask me about teaching themselves to "code". 1. Find a thing you want to make. 2. Find out how to make it. 3. Try to make it. 4. Learn the skills that previously prevented you from making it.
- 13415 8y ago
  I don't think this is a good comparison. People who are skilled in both areas might not realize that, but learning how to program is a hundred times easier than working through any advanced topic in higher mathematics and coming up with your own proofs.
  - jcora 8y ago
    This thread seems bizarre to me. There's one guy claiming he "worked himself up to graduate level math" in the last two years of high-school. So either he's a literal IMO-gold-medal-level genius, or people here don't quite understand what undergraduate math studies actually involve. Even if he is that intelligent there's no way he actually did the amount of work required on such a broad number of topics.
    - oefrha 8y ago
      Can confirm. I’m not an IMO gold medalist, but I come from a very competitive nation that fetch five or more gold medals on most years, and I almost made the national team, twice (failed at TST, which selects 6 out of ~30). And I graduated with a math degree from one of the top institutions. There’s no way I would have completed undergrad plus entry level grad math in the last two years of high school — those took me three years in college (of course I was doing other things, but still). Unlike programming, mathematics isn’t something you can pick up in a weekend. EDIT: Now that I think about it, you can probably bang your head against, say, Lang, for two years, “digesting” a big chunk of it, earning you the bragging right of “working yourself up to graduate level math”. That won’t give you the breadth of a good bachelor of mathematics, and it certainly won’t prepare you for quality research.
  - 8y ago
deleted 8y ago
[deleted]
proginthebox 8y ago
This is terrible advice. Apart from the last sentence. A better advice would be to specify which subject to learn. For example, (since I don't really have much time) 1. Topology (book by Munkres) 2. Real Analysis and Measure Theory (book series by Stein Shakarchi) 3. Algebra (book by Aluffi) 4. Linear Algebra (book by Friedberg Insel) 5. Measure Theoretic Probability (book by Cinlar) 6. Differential Geometry (book Smooth Manifolds by Lee) 7. Numerical Analysis (book by Quarteroni) 8. Set Theory and Propositional Logic (books by Goldrei) This is what one will mainly learn in a strong undergrad/grad math program. Once this is done, then there are different tracks to follow.
- abnry 8y ago
  The advice I gave is not exclusive to working through the typical undergraduate books. I was questioning why the OP wants to self-study an undergrad math curriculum to begin with. It's probably not to become a pure mathematician. So I suggested, instead of creating a massive goal of getting through a collection of books just for the sake of being a completionist, to have a concrete personal goal. Otherwise, people can throw books "you have to read" at you until the cows come home. Especially since this person is talking about applied math.
  - proginthebox 8y ago
    I see. Well, you have a point, but the OP did specifically ask for a plan like that Susan Fowler's blog post. And I am an applied mathematician working mostly on computational physics and I can attest to the requirements I mentioned. But your advice has a point, just going through books mindlessly is not motivation enough/ can lead to wandering. And it is always good to have specific tasks at hand. Like, solving a particular ordinary differential equations numerically.
- xenihn 8y ago
  Is this the recommended order to read/learn them in? If I was primarily concerned with getting up to par with math for the sake of being able to actually understand everyone's favorite algorithms textbooks, would you still suggest working through all of these, or could you recommend an abridged list?

by abnry 8y ago

foo101 8y ago

Is this sentence incomplete or am I missing the context of the word "compelling" here? > be able to write up a compelling Can someone clarify what this means?

mindcrime 8y ago
I suspect it was meant to say "be able to write up a compelling proof".
abnry 8y ago
By mistake I swapped lines in my post when composing it. It is fixed now.

joe_the_user 8y ago

The OP did specify what he wanted - the basic undergraduate and starting graduate curriculum. That's a pretty well defined area: Algebra, Real Analysis, Geometry and Topology with maybe some complex analysis, number theory, statistics, CS or etc thrown in. I personally did work myself up to the graduate in math during the last two years of High School & first year of college. I was motivated by exploring ideas and gaining knowledge. I would guess that each person has a somewhat unique motivation strategy. Maybe solving problems gets some people doing stuff. I'm sure simply learning stuff can motivate others. Probably each person has to experiment to discover what works for them - I would pick up a calculus book and read it - well, I'd skim repeatedly and then read in depth, solving a few problems. Math is difficult, of course, so having a bit of patience with your until it gets the ideas on it's own is probably necessary.

throwawaymath 8y ago
> I personally did work myself up to the graduate in math during the last two years of High School & first year of college. ...how in the world did you manage to do this? Did you actually self-study, or were you placed in a gifted program? Self-studying all of undergraduate mathematics is more impressive than actually studying all of it in a four year classroom setting. Doing so as a teenager is amazing. The most gifted person I've ever known in mathematics is actually a physicist who entered Harvard at 16. He was already taking undergraduate math courses at 13/14 and had mastered all the undergraduate material by the first semester of his undergraduate degree. For the remaining years of his undergraduate degree he took graduate math courses. But in order to do that he needed to not only be gifted; he was placed in a program for extremely gifted kids sponsored by Stanford University from his preteen years. I don't think your anecdote is a great comparison for the OP's expectations (or for calibrating advice they'd benefit from).
- joe_the_user 8y ago
  How in the world did you manage to do this? Did you actually self-study, or were you placed in a gifted program? I read quite a bit on my own, I took some courses at UCLA through a high school scholars program (including the undergraduate honors seminar). The entirety of the undergraduate program might be a slight exaggeration but I was ready for graduate level courses when I got to Berkeley. I think going through the material requires determination, not necessarily being extremely gifted. But then, it seems like people at someone's gone through a bunch material say by that fact they're gifted. Thus having done this, one is tautologically gifted. I've never tested at the extremely gifted level but I'm doubtful of single-measures of intelligence regardless. Edit: I don't think your anecdote is a great comparison for the OP's expectations (or for calibrating advice they'd benefit from). Neither of us know the OP. It's kind of up to them to calibrate what process works for them. Scanning a lot of math until I found good, clear explanations worked well for me.
analog31 8y ago
>>> I would guess that each person has a somewhat unique motivation strategy. Indeed, speaking only for myself, I don't think that I had the self discipline to carry through with such a plan as a teenager. I needed the classroom environment to learn math & physics, which became my college majors. At the same time, I was able to teach myself programming and electronics quite easily for some mysterious reason. And I ended up combining all of those things in grad school. Also, I only had access to the books and resources of a typical suburb with a decidedly anti-intellectual culture. The officials at my high school refused to offer a calculus course because they said it would be "elitist."

forkandwait 8y ago

Personally, I think this is bad advice, because without an undergrad+ background, the projects above will either be impossibly frustrating or you will make up some crackpot bullshit. Plus, the undergraduate curriculum is its own reward.

enraged_camel 8y ago
Nah, a good way to learn new skills is to pick a destination and then figure out what steps you need to take to get there. This type of “top-down” learning can help one stay motivated through the most frustrating road blocks. This is especially important for self-learning, because unlike an undergrad setting, the person is on their own and can’t rely on peers.
- forkandwait 8y ago
  Well, we disagree. Plus, taking on some crazy problem without background saps motivation as well.
  - ny8jjang 8y ago
    I think setting attainable goals and working towards it helps. It is very easy to lose motivation if you don't know what you are going for..
    - forkandwait 8y ago
      Depending on your background, you probably don't have enough information to pick a long-term goal anyway. I am afraid that sounds curmudgeonly, but I have also seen students shoot themselves in the foot because they decided they didn't need a class for their not very well informed goals.
      - enraged_camel 8y ago
        >>Depending on your background, you probably don't have enough information to pick a long-term goal anyway. Nah, it's totally possible for newbies to pick high-level long-term goals. This can be something like "I want to teach my computer to tell apart dogs and cats", or "I want to create a website where people can buy and sell yarn." From there, Google searches can direct someone towards concepts and various methods of learning them. I mean, you can disagree all you want, but this is in fact how many people learn things.

madhadron 8y ago

This is really good advice, and I thinking about suggesting the same thing. I've accepted that there are some basics that you have to have in place before you can really get traction and learn what you need to learn for specific projects.

40acres 8y ago

I think this is good advice. I tried to read parts of CLRS a few years ago but could not wrap my head around the work. The mathematical prerequisites needed for CLRS gave me an end point to guide my self learning to focus on specific areas: algebra (linear and elementary), calculus 1, combinatorics / probability, discrete math, propositional logic.

te_platt 8y ago

My oldest brother had a real love of mathematics and got a masters degree but never even applied to a PhD program. When I asked him about it he just said he was tired of working on other people's problems. He had a quiet programming position in a large company that gave him time to work on personal projects. Among other things he volunteered at the local schools teaching "fun" math. I guess I'm just agreeing with you about finding something that inspires you and go with it.

selimthegrim 8y ago
I don’t know what PhD program your brother was looking at but ideally candidacy committees won’t pass you unless you chose the problem or made it your own. Where your brother is right is that forming such a ideal committee is mostly politics.
- Eridrus 8y ago
  I don't know about math, but in CS, a lot of funding comes from specific grants, so you'll be working on something related to what your advisor got grant funding for.
  - QuercusMax 8y ago
    Regardless of grants, etc., your general area of work is going to be based on what the faculty members' focuses are. But in any case, you'll probably still be doing your own sub-piece of some larger effort. You wouldn't choose an advisor at random, so this isn't really relevant. I mean, if you're interested in astrophysics, you shouldn't go to a school where the Physics department is focused on nuclear physics!
- te_platt 8y ago
  I think you are right on both points. There were more details and his answer was just a quick way to say: He didn't care about the status of having a PhD He didn't need a PhD to advance his career He was having fun doing math stuff anyway
8y ago

MattHeard 8y ago

This is exactly the kind of advice I give to people who ask me about teaching themselves to "code". 1. Find a thing you want to make. 2. Find out how to make it. 3. Try to make it. 4. Learn the skills that previously prevented you from making it.

13415 8y ago
I don't think this is a good comparison. People who are skilled in both areas might not realize that, but learning how to program is a hundred times easier than working through any advanced topic in higher mathematics and coming up with your own proofs.
- jcora 8y ago
  This thread seems bizarre to me. There's one guy claiming he "worked himself up to graduate level math" in the last two years of high-school. So either he's a literal IMO-gold-medal-level genius, or people here don't quite understand what undergraduate math studies actually involve. Even if he is that intelligent there's no way he actually did the amount of work required on such a broad number of topics.
  - oefrha 8y ago
    Can confirm. I’m not an IMO gold medalist, but I come from a very competitive nation that fetch five or more gold medals on most years, and I almost made the national team, twice (failed at TST, which selects 6 out of ~30). And I graduated with a math degree from one of the top institutions. There’s no way I would have completed undergrad plus entry level grad math in the last two years of high school — those took me three years in college (of course I was doing other things, but still). Unlike programming, mathematics isn’t something you can pick up in a weekend. EDIT: Now that I think about it, you can probably bang your head against, say, Lang, for two years, “digesting” a big chunk of it, earning you the bragging right of “working yourself up to graduate level math”. That won’t give you the breadth of a good bachelor of mathematics, and it certainly won’t prepare you for quality research.
- 8y ago

deleted 8y ago

[deleted]

proginthebox 8y ago

This is terrible advice. Apart from the last sentence. A better advice would be to specify which subject to learn. For example, (since I don't really have much time) 1. Topology (book by Munkres) 2. Real Analysis and Measure Theory (book series by Stein Shakarchi) 3. Algebra (book by Aluffi) 4. Linear Algebra (book by Friedberg Insel) 5. Measure Theoretic Probability (book by Cinlar) 6. Differential Geometry (book Smooth Manifolds by Lee) 7. Numerical Analysis (book by Quarteroni) 8. Set Theory and Propositional Logic (books by Goldrei) This is what one will mainly learn in a strong undergrad/grad math program. Once this is done, then there are different tracks to follow.

abnry 8y ago
The advice I gave is not exclusive to working through the typical undergraduate books. I was questioning why the OP wants to self-study an undergrad math curriculum to begin with. It's probably not to become a pure mathematician. So I suggested, instead of creating a massive goal of getting through a collection of books just for the sake of being a completionist, to have a concrete personal goal. Otherwise, people can throw books "you have to read" at you until the cows come home. Especially since this person is talking about applied math.
- proginthebox 8y ago
  I see. Well, you have a point, but the OP did specifically ask for a plan like that Susan Fowler's blog post. And I am an applied mathematician working mostly on computational physics and I can attest to the requirements I mentioned. But your advice has a point, just going through books mindlessly is not motivation enough/ can lead to wandering. And it is always good to have specific tasks at hand. Like, solving a particular ordinary differential equations numerically.
xenihn 8y ago
Is this the recommended order to read/learn them in? If I was primarily concerned with getting up to par with math for the sake of being able to actually understand everyone's favorite algorithms textbooks, would you still suggest working through all of these, or could you recommend an abridged list?