I have a pretty similar background. I have an undergrad in ChemE who fell in love with machine learning research. As I didn’t had the appropriate background so I taught myself Computer science using mostly resources such as OCW and teachyourselfcs, videolectures etc.
However, what stood out to me was how difficult it is to self study? Universities provide a setting which helps you learn difficult subjects over a longer period of time. Outside of that, no such avenues exist. It’s not just reading up a book or making Anki flash cards(which is quite tedious to be honest) but the process of selecting, vetting what to read next and actually completing it. Or the question are we there yet, with no idea what “there” actually means.
I decided to work on creating a platform where people can actually learn difficult subjects on their own. It’s a bit different than normal video based platform, it actually uses text based conversation to facilitate learning. But I truly believe this is the way forward.
I have written comic based guide about this problem:
I would love to self study with more vigor. I studied Mechanical engineering, but I find Math to be so beautiful. I think I chose the right education for the work I find meaningful in the world, but I wish I had more time to learn math that isn't directly relevant to what I am doing.
I'll certainly check out your site and save it for when the world is more peaceful.
> However, what stood out to me was how difficult it is to self study? Universities provide a setting which helps you learn difficult subjects over a longer period of time. Outside of that, no such avenues exist. It’s not just reading up a book or making Anki flash cards(which is quite tedious to be honest) but the process of selecting, vetting what to read next and actually completing it.
One major thing I found when learning at a university that I have trouble with when self studying is pacing.
Let's say the textbook covers some topics A, B, C, D, and E in that order. When self studying I start with A, and keep working on A until I'm confident and comfortable with it, then move on to B, and so on.
In school we'd spend a while on A, but then usually move on to B before I was fully comfortable and confident with A. Same for B to C, and so on.
This was fine, because while I didn't think I was ready to move on the instructors had a lot of experience with teaching this material and knew that we students knew enough A to understand B, and in fact that exercising our A to help us learn B would also strengthen our A.
Excellent intro comic, it got me really excited about your product! I would love to try it out once you release a more advanced course. I signed up, hopefully you will send e-mail updates as you release new courses.
This is very well done. I've been hobby researching learning tools for the past couple years and this looks like one of the best. @ai_ia have you seen Andy Matuschak's Orbit project?
I self study machine learning here https://learnaifromscratch.github.io/ai.html it's an early and shitty draft and proof of concept that you can do self-directed learning for these topics while looking up the background you need to know, which for me is much more interesting than taking a generalized math curriculum of absolutely everything. The courses so far we haven't escaped the content of Wasserman's 'All of Statistics' book yet on classification or probabilistic graphs, so you could if you wanted watch the lectures and only do Wasserman's book.
If you want to try the OCW linear route of taking everything for whatever reasons, you will have to get up to MIT student levels trying to unravel the algebra done in the early calculus courses and later where they just assume you possess this background. One way to do that is those problem solving books like this one which 'bridges the gap between highschool math and university' https://bookstore.ams.org/mcl-25 at least you then get worked out solutions. Another way is Poh-Shen Loh's Discrete Math course he opened up on YouTube which is done the same way he holds the CMU Putnam seminar, working through a bunch of combinatorics and algebra will more than prepare you to understand those continuous math OCW courses https://youtu.be/0K540qqyJJU
Like everybody else there is of course the issue of: who is going to check my work. For me I went with the time tested tradition of hiring a tutor, a local grad student and paid them once a week to go on chat/zoom or meet at a coffee shop before the pandemic and spend a few minutes going over everything I'm doing wrong. In the early days however I used constructive logic ie: 'proof theory', to audit my own work: https://symbolaris.com/course/constlog-schedule.html and read a huge amount of Per-Martin Lof papers on the justifications of logical operators like implies, disjunction, conjunction, etc. Of all the math I've ever taken I would say that proof theory was the most useful for somebody by themselves who isn't sure of what they are doing (I'm still not 100% sure.. hence why I hire people now).
If you want a great Calculus text that explains those nasty looking Euler's e nested statistics distributions try Mathematical Modeling and Applied Calculus by Joel Kilty everything from partial derivatives, gradients, x^n, e^x, trig, integrals, limits is explained in terms of parameters to modeling functions, if you write software it will be easy to understand. I haven't posted it yet but I tried going through Allan Gut's probability book using only that math modeling calc text and have not run into anything applied, as in concepts about limits or integrals, that wasn't already covered. Of course the concepts are much more abstract measuring a bunch of intervals and a different method of integration and I don't pretend I'll be making any advances in this area beyond applied usage but it can be done, jump in and pick up the background as you go as opposed to doing all the background at once, losing interest and giving up.
If anyone is interested in following a self-study curriculum but with more of an emphasis on pure vs applied math, you may find my experience and recommendations useful:
I’m doing something similar, except for Stats. I’ve cobbled together a plan based on degree programs from Stanford, CMU, and Berkeley. It would seem easier to stay on track with directed course learning, but how do you stay on track with the self-directed learning?
Is this doable for someone with less basis in math? I almost stopped studying math after two years of undergrad, when I went to a more practically-focused school. It'd be nice to get back into shape, because I've started reading research articles and always feel like I'm behind on the theory side of ML.
It depends very much on your own skills and drive.
At some point, you need people to check your work, though, but you can search for online communities to do that. E.g. a post to mathoverflow asking about whether the following proof is correct, for example. Try to find a community of others online and work with them.
Also remember that math is about ideas. It is not just an exercise in formal deduction. The field is dense with non-trivial, non-obvious, interesting insights, and understanding these insights so well that overtime they become obvious to you is what it means to learn math. Sites like 3Blue1Brown to a good job trying to explain these ideas, but they really put a lot of work into it. Most math texts do not, but you have a professor or classmate you can talk to -- so there is a gap between the text and what you need, but perhaps you can close the gap with online resources.
The issues will be the same with self studying anything. Without someone to critique your work, and point out deficiencies you don't even know are a thing, continuous improvement quickly becomes exponentially harder from the bad habits that are holding you back.
If anyone wants to attempt this, that's awesome. It's a lot of hard work, but I think it opens a lot of opportunities for personally and professionally. I've a Ph.D in Applied Mathematics from a traditional program, so I wanted to chime in based on some of the comments I'm seeing. As a note, this is an opinion and others may feel differently and strongly at that.
To me, applied mathematics is the art of transforming something into a linear system, which is something that we can tangibly solve on a computer. There are lot's of ways to do this such as Taylor series and Galerkin methods, so a lot of the field is understanding how, when, and why each method can be used. This is coupled with mastery over linear solvers, which includes direct methods, iterative methods, preconditioners, etc.
I wanted to write this comment, though, to focus on certain areas that may end up blocking what I view as appropriate progression in the field. These are things that I believe are necessary to understand advanced topics, but don't necessarily fall under applied mathematics. First, you really do need mastery of real analysis. It's necessary because it covers formally topics such as differentiation, integration, and series, which are required to understand theorems and algorithms. In my opinion, calculus books are not sufficient. Rudin's Principle's of Mathematical Analysis is the most concise, well written book that contains enough. Second, enough functional analysis to understand Hilbert Spaces is required. This prerequisite to this is the real analysis above. The issue here is that algorithms for things like differential equations require function spaces to do properly. Certainly, you can go really deep in this regard, but Hilbert Spaces are generally enough for practical algorithms. This also affects optimization theory, which impacts machine learning. Technically, you can do optimization theory with only real analysis, but the theory is cleaner in Hilbert Space. Questions that need to be answered are things like does the infimum exist and can it be obtained? Working with a general inner product is also a valuable tool for parallelization as well as a modeling tool. Third, some integration or measure theory is required. It depends on what you're doing, so I don't think mastery is strictly necessary, but spaces like L2 don't make a lot of sense unless you know what a Lebesgue integral is. Even if you want to just work with spaces that are Riemann integrable, measure theory helps understand when this is possible and the ramifications of it. And, to be clear, this is important outside of differential equations. If you want to understand optimization theory in a Hilbert Space, the inner products used will require some understand of measure theory.
Anyway, these are some random thoughts and ideas about the field. I do believe strongly that any amount of study is beneficial as most engineering fields benefit from applied mathematics.
As far as books go, we used "Real Analysis" by Carothers for analysis in undergrad at Caltech and it's one of only two math textbooks (the other being Dummit and Foote for algebra) that I go out of my way to recommend, particularly for self study. It's probably not a sufficient book if you're pursuing a Ph.D. or anything, but I would definitely not start out with Rudin (too little hand-holding, you'll die), and Carothers was amazing.
I tried to do the honors calc sequence through OCW and I found that I really couldn't hack it on the textbook alone, I'd get ~60% on my self-graded homeworks and I'd spend weeks on individual problems. Often I'd get stuck just trying to figure out the notation, which differed from the book. And I found that the recorded lecture material runs out pretty quick beyond intro level classes.
I ended up enrolling in the math 23 sequence at the Harvard Extension School and found it to be a lot more valuable.
Just don't spread yourself out too thin, and know when to stop - it's super easy to burn out when taking on too much coursework, even if this extra part is self study.
The author writes that he wants to pursue a Ph.D - in that case, I'd put all my effort into pursuing what would maximize my chances at getting into such program. Unfortunately, self study is quite difficult to prove, and does not hold much weight when applying for such positions.
This guy was my TA in a GT for a course on Educational Technology. He totally blew off giving me any feedback until the last possible moment, super frustrating experience.
[+] [-] ai_ia|4 years ago|reply
However, what stood out to me was how difficult it is to self study? Universities provide a setting which helps you learn difficult subjects over a longer period of time. Outside of that, no such avenues exist. It’s not just reading up a book or making Anki flash cards(which is quite tedious to be honest) but the process of selecting, vetting what to read next and actually completing it. Or the question are we there yet, with no idea what “there” actually means.
I decided to work on creating a platform where people can actually learn difficult subjects on their own. It’s a bit different than normal video based platform, it actually uses text based conversation to facilitate learning. But I truly believe this is the way forward.
I have written comic based guide about this problem:
https://primerlabs.io/comics/introducing-primer-comics/
We are creating self paced courses on Computer Science and also looking into creating self paced Mathematics/Physics courses as well.
We have released two free courses for everyone to try out at https://primerlabs.io
[+] [-] WhisperingShiba|4 years ago|reply
I'll certainly check out your site and save it for when the world is more peaceful.
[+] [-] tzs|4 years ago|reply
One major thing I found when learning at a university that I have trouble with when self studying is pacing.
Let's say the textbook covers some topics A, B, C, D, and E in that order. When self studying I start with A, and keep working on A until I'm confident and comfortable with it, then move on to B, and so on.
In school we'd spend a while on A, but then usually move on to B before I was fully comfortable and confident with A. Same for B to C, and so on.
This was fine, because while I didn't think I was ready to move on the instructors had a lot of experience with teaching this material and knew that we students knew enough A to understand B, and in fact that exercising our A to help us learn B would also strengthen our A.
[+] [-] maddyboo|4 years ago|reply
[+] [-] vertak|4 years ago|reply
https://withorbit.com/
You both seem to be solving similar problems.
[+] [-] flavio_x|4 years ago|reply
[+] [-] Evgenii1|4 years ago|reply
If you want to try the OCW linear route of taking everything for whatever reasons, you will have to get up to MIT student levels trying to unravel the algebra done in the early calculus courses and later where they just assume you possess this background. One way to do that is those problem solving books like this one which 'bridges the gap between highschool math and university' https://bookstore.ams.org/mcl-25 at least you then get worked out solutions. Another way is Poh-Shen Loh's Discrete Math course he opened up on YouTube which is done the same way he holds the CMU Putnam seminar, working through a bunch of combinatorics and algebra will more than prepare you to understand those continuous math OCW courses https://youtu.be/0K540qqyJJU
Like everybody else there is of course the issue of: who is going to check my work. For me I went with the time tested tradition of hiring a tutor, a local grad student and paid them once a week to go on chat/zoom or meet at a coffee shop before the pandemic and spend a few minutes going over everything I'm doing wrong. In the early days however I used constructive logic ie: 'proof theory', to audit my own work: https://symbolaris.com/course/constlog-schedule.html and read a huge amount of Per-Martin Lof papers on the justifications of logical operators like implies, disjunction, conjunction, etc. Of all the math I've ever taken I would say that proof theory was the most useful for somebody by themselves who isn't sure of what they are doing (I'm still not 100% sure.. hence why I hire people now).
If you want a great Calculus text that explains those nasty looking Euler's e nested statistics distributions try Mathematical Modeling and Applied Calculus by Joel Kilty everything from partial derivatives, gradients, x^n, e^x, trig, integrals, limits is explained in terms of parameters to modeling functions, if you write software it will be easy to understand. I haven't posted it yet but I tried going through Allan Gut's probability book using only that math modeling calc text and have not run into anything applied, as in concepts about limits or integrals, that wasn't already covered. Of course the concepts are much more abstract measuring a bunch of intervals and a different method of integration and I don't pretend I'll be making any advances in this area beyond applied usage but it can be done, jump in and pick up the background as you go as opposed to doing all the background at once, losing interest and giving up.
[+] [-] nsainsbury|4 years ago|reply
https://www.neilwithdata.com/mathematics-self-learner
[+] [-] blparker|4 years ago|reply
[+] [-] querez|4 years ago|reply
[+] [-] unknown|4 years ago|reply
[deleted]
[+] [-] joconde|4 years ago|reply
[+] [-] rsj_hn|4 years ago|reply
At some point, you need people to check your work, though, but you can search for online communities to do that. E.g. a post to mathoverflow asking about whether the following proof is correct, for example. Try to find a community of others online and work with them.
Also remember that math is about ideas. It is not just an exercise in formal deduction. The field is dense with non-trivial, non-obvious, interesting insights, and understanding these insights so well that overtime they become obvious to you is what it means to learn math. Sites like 3Blue1Brown to a good job trying to explain these ideas, but they really put a lot of work into it. Most math texts do not, but you have a professor or classmate you can talk to -- so there is a gap between the text and what you need, but perhaps you can close the gap with online resources.
[+] [-] zsmi|4 years ago|reply
The issues will be the same with self studying anything. Without someone to critique your work, and point out deficiencies you don't even know are a thing, continuous improvement quickly becomes exponentially harder from the bad habits that are holding you back.
Technique totally matters, even in STEM.
[+] [-] downrightmike|4 years ago|reply
[+] [-] kxyvr|4 years ago|reply
To me, applied mathematics is the art of transforming something into a linear system, which is something that we can tangibly solve on a computer. There are lot's of ways to do this such as Taylor series and Galerkin methods, so a lot of the field is understanding how, when, and why each method can be used. This is coupled with mastery over linear solvers, which includes direct methods, iterative methods, preconditioners, etc.
I wanted to write this comment, though, to focus on certain areas that may end up blocking what I view as appropriate progression in the field. These are things that I believe are necessary to understand advanced topics, but don't necessarily fall under applied mathematics. First, you really do need mastery of real analysis. It's necessary because it covers formally topics such as differentiation, integration, and series, which are required to understand theorems and algorithms. In my opinion, calculus books are not sufficient. Rudin's Principle's of Mathematical Analysis is the most concise, well written book that contains enough. Second, enough functional analysis to understand Hilbert Spaces is required. This prerequisite to this is the real analysis above. The issue here is that algorithms for things like differential equations require function spaces to do properly. Certainly, you can go really deep in this regard, but Hilbert Spaces are generally enough for practical algorithms. This also affects optimization theory, which impacts machine learning. Technically, you can do optimization theory with only real analysis, but the theory is cleaner in Hilbert Space. Questions that need to be answered are things like does the infimum exist and can it be obtained? Working with a general inner product is also a valuable tool for parallelization as well as a modeling tool. Third, some integration or measure theory is required. It depends on what you're doing, so I don't think mastery is strictly necessary, but spaces like L2 don't make a lot of sense unless you know what a Lebesgue integral is. Even if you want to just work with spaces that are Riemann integrable, measure theory helps understand when this is possible and the ramifications of it. And, to be clear, this is important outside of differential equations. If you want to understand optimization theory in a Hilbert Space, the inner products used will require some understand of measure theory.
Anyway, these are some random thoughts and ideas about the field. I do believe strongly that any amount of study is beneficial as most engineering fields benefit from applied mathematics.
[+] [-] RheingoldRiver|4 years ago|reply
[+] [-] bigdict|4 years ago|reply
Thanks for the insightful comment by the way.
[+] [-] capitalsigma|4 years ago|reply
I ended up enrolling in the math 23 sequence at the Harvard Extension School and found it to be a lot more valuable.
[+] [-] TrackerFF|4 years ago|reply
The author writes that he wants to pursue a Ph.D - in that case, I'd put all my effort into pursuing what would maximize my chances at getting into such program. Unfortunately, self study is quite difficult to prove, and does not hold much weight when applying for such positions.
[+] [-] andi999|4 years ago|reply
[+] [-] bigdict|4 years ago|reply
[+] [-] tgflynn|4 years ago|reply
[+] [-] adnmcq999|4 years ago|reply
[+] [-] unknown|4 years ago|reply
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[+] [-] toreply2021|4 years ago|reply
[deleted]
[+] [-] wespiser_2018|4 years ago|reply
I guess now I know where his time went?
[+] [-] unknown|4 years ago|reply
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[+] [-] toreply2021|4 years ago|reply
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[+] [-] jchristian-|4 years ago|reply
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[+] [-] CamperBob2|4 years ago|reply