I have a BSc in computer engineering and part of the major was just a couple of calculus and discrete math courses. I want to expand my knowledge in mathematics through online courses. Is there a specialized website that offers courses in various mathematics fields?
I am a math professor. In my observation, there is a huge amount of material available on the web, but it isn't very centralized -- especially at the upper levels.
My advice would be to get a book on any topic which interests you, read through it, and do a significant number of the exercises. You might try Epp's Discrete Mathematics, Hefferon's Linear Algebra, Colley's Vector Calculus, Dudley's Elementary Number Theory, Spivak or Apostol's Calculus (these go far beyond ordinary freshman calculus), Pinter's A Book of Abstract Algebra, among many others. Some of these books are expensive to buy new, but just buy older editions.
Resources like Khan Academy and 3blue1brown are also fantastic, and I have shared some of these with my students. I'd recommend using these as a supplement; if you rely on them solely then you'll develop vague intuition but not much else.
Also, with the pandemic, there are a huge number of traditional university courses that have moved online, and you could probably enroll in one for not too much money. Check the RateMyProfessor reviews -- you want a mix of positive reviews claiming the prof inspired them, and negative reviews complaining that work was expected. I have a RateMyProfessor review which complains bitterly that "homework is graded for accuracy and not completion". :)
With many books, I'm not sure how to verify that I'm actually doing the exercises properly.
In the (simple!) maths I took, my misunderstandings often led me to do exercises incorrectly, but to think I'd done them right, or to have half a proof and no idea what insight I was missing. Even when solutions were provided (which was rare!), the path connecting the dots could be foggy. "What made them think to introduce this lemma?" kind of questions.
I often needed discussion with others to help figure out my misconceptions and correct them -- so... big ask, but do you have any further suggestions on how to replace talking with a TA, or with a smart group of upper-year friends?
> I'm not sure how to verify that I'm actually doing the exercises properly.
Since the parent mentioned my text, Linear Algebra, I'll note that it is Free, and comes with worked answers to all exercises, including proofs. And since other posters mention videos, I'll also note that there are videos (and the slides are available), along with a lab manual using Sage.
https://hefferon.net/linearalgebra/index.html
I'd recommend not only asking, but also answering others' questions. It's not the same as having a group of people that you regularly talk to, but it's something.
You could also hire a graduate student tutor. For example, here is a list of math grad students at the University of Wisconsin willing to do this:
Another alternative to hanging out in forums is to join a good math discord server. I can highly recommend this one: https://discordnetwork.com/, I have no idea how many people are in the server but I'm guessing at least more than 10k (about 9k online as I'm writing this, and more offline I have to presume). The advantage with such servers is fluid communication, but the downside is because of this in combination with the fact that there might be alot of people, you sometimes have to wait as to not disturb ongoing discussions.
This discord server in particular is good if you want help, although you might not recieve instant help, you will eventually get help if you are patient. There are also other channels in the server just for the purpose of discussion, i.e discussing general topics in math.
This might not give you what you are looking for, but it's a possible alternative I guess.
Another method for checking your understanding is to work through the worked out problems in the book and any proofs it may have, too. (Most of the value in a proof is in the techniques used to prove the result.) Work through them yourself, trying to understand each step, and attempt to do the problems in the same way. At first, justify each step you take, don't try to skip the "obvious" steps. Again, this isn't a perfect substitute for an instructor, but it can go a long way towards your goal.
I've self-taught myself undergraduate-level Mathematics, and am currently enrolled in an advanced Masters program specialized in higher category theory at Université de Paris. Here are some thoughts:
First, the amount I learnt on a day-to-day basis from self-studying in my free time pales in comparison to the amount I learnt from a structured course ending with an exam, in which you have to commit to studying 2~4 hours of math every single day.
Second, classroom lectures mainly act as a structured table-of-contents for working through various textbooks. The problem sheets they distribute in the tutorials are very helpful, as is the discussion of problems that some students might have got stuck in.
Graduate-level math textbooks are fantastic, and the authors have poured 10+ years of their lives writing them. No online course with cute interactive video can substitute them. Having said that, there are some video lectures that can act as good motivation for studying a subject. For instance, Wildberger's lectures on algebraic topology (find them on YouTube) are a great starting point for the subject.
You first have to decide on what you want to learn, by looking through various fields, and narrow your scope. Mathematics is a huge discipline, and you have to decide what you like, and work through the prerequisites in a disciplined manner.
I cannot emphasize the importance of working through exercises enough. All mathematics textbooks have exercises at the end of each section, a subset of which you must work through.
Having said all this, you might find the following site interesting: it contains a lot of unpolished notes from studying various math textbooks, but it requires a lot more work to be useful to a larger audience. Look through it, and pick something you like. On request, I might find the time to add some material.
I always appreciate when someone leaves the well-defined path and does something different, such as you self-studying Math and getting enrolled in a Masters program.
I am just curious about your process of actually joining the masters program. They usually require 2-3 letters of recommendation etc. How did you handle such requirements?
> Second, classroom lectures mainly act as a structured table-of-contents for working through various textbooks.
On a side note, actually studying math as an undergrad, I can't stress enough how much I agree with this, especially in a remote setup.
Can you speak towards what you self-studied to get to an undergraduate understanding of mathematics and what resources, or books you used in the process? I’m also interested in the resources you used or recommend for learning category theory.
Motivation and direction are important when starting out. I decided pretty early on that I wanted to be an algebraist, but would have to build some mathematical maturity before I could get there, so I had a rather shallow goal in the beginning: to be able to solve the previous years' math GRE papers. Off the top of my head, these were some of the books that I worked through:
1. Spivak's Calculus.
2. Johnstone's Notes on Set Theory and Logic.
3. Gamelin's Complex Analysis.
4. Hoffman & Kunz' Linear Algebra.
5. Dummit & Foote's Abstract Algebra; just the group theory.
6. Munkres' Topology; just the general topology.
Once I was happy with my preparation, I strived for a deeper understanding of Group Theory. I bumbled through Herstein, but didn't understand it very well. Then, I stumbled upon Artin's book, and worked through it using the outline provided on the MIT OCW course page, and I could confidently solve most of the exercises.
For category theory, the top resources that I would recommend are:
1. Mileweski's Category Theory for Programmers, the video lecture series. As is always the case with video lectures, this one can help motivate a Haskell programmer uninitiated in category theory.
2. Goldblatt's Topoi. It's fairly dated, but teaches category theory well, via its application to topoi.
3. MacLane's CatWork. I'm not especially fond of this one, but it's necessary to work through it.
The most important thing to understand when learning category theory is that it cannot be learnt in a vacuum: the subject is entirely vacuous, and you need to use it in other mathematical disciplines to give it meaning.
I started in late 2016. Initially, it was just during the weekends. I was a regular at the local coffee shop near my place in Boston: I'd go in with my iPad and some sheets of paper, spend 4~5 hours studying, while constantly consuming coffee. Then, I found a friend who had done a Bachelors in math, and we'd visit the local community college 3~4 times a week, and discuss general math and GRE problems on the blackboard for a couple of hours after work: it was a lot of fun.
In mid-2017, I moved to California and spent ~2 hours a day studying by myself, while maintaining a day job writing Haskell and Coq. Then, in late-2019, I quit my job and moved to India to study mathematics full-time (read: 4~6 hours every single day). I started meeting a professor at the local university once a week to discuss my solutions to problems in Miles Reid. We worked through Reid together. I also audited a course in algebraic topology at the university, simultaneously.
In mid-2019, I moved to Paris to continue studying part-time (1~2 hours a day) while working on a Coq project. After some shuffles, I found a professor I really liked, and we started working together. He wrote me the primary recommendation for the Masters program.
I hope the elaboration was more useful than two numbers.
If I might ask - which local university was it that let you audit their course? This is not a usual occurrence in India, which is why I ask -- I'm extremely happy to hear that things like this even happen as outliers, btw.
The fact that it was a graduate-level course in algebraic topology seems to imply that the university in question is not a run-of-the-mill university, which makes me all the more happy.
I would advice against resources like KA and 3B1brown.
I'm a firm believer that you don't truly understand material unless you also understand a proof for it. Ane while those resources are AMAZING for simple intuition building, they aren't sufficient.
Go pick up a book about proofs to develop mathematical maturity and then delve into some undergrad books for analysis (Abbott) or linear algera (Axler)
I would advise the opposite, especially 3 blue 1 brown. I think getting the intuition first before diving into the deep proofs gives you a north star so you know where proofs will be taking you. I believe that for most people, getting the intuition first is going to make learning a lot easier.
Absolutely you should not use 3B1B as a replacement for reading a textbook and doing problems.
However, I don’t think you that means you should skip watching 3B1B videos at all. In fact, Grant repeats quite often in his videos that they’re not a substitute for doing problems.
I’d highly recommend watching the videos side by side with reading through Axler. They will help develop the right geometric intuition (which Axler does a decent job of, but books can only do so much as a medium). I believe the 3B1B videos can also be quite motivating, which in my experience is the most important factor when self studying.
Perhaps another facet to this is that both KA and 3B1B are primarily video lectures, and it's common knowledge that simply listening to someone explain something performs poorly for learning relative to actually getting hands-on with the topic. KA is even trying to fill this gap. They've added interactive visualizations for some topics.
I'd say the same is true for proofs. Simply reading a proof isn't the best if you want to understand/retain it. If you can "re-invent" the proof step by step (or in some contexts you might be able to translate steps between geometric and algebraic representations), then you have probably internalized it sufficiently.
Yep. I love watching 3B1brown, but those seem to be in-depth videos on a particular topic.
I was wondering how you feel about Professor Leonard? He's pushing recordings of his college classes online on YouTube. It's a veritable trove of useful lectures. [1]
Axler is too abstract for most people who want applications (I have worked through it). Unfortunately, I don't know of a good linear algebra text that is focused on actual calculations and isn't boring. Linear algebra has so many interesting applications, but I found learning it tedious.
While I agree they are not enough to build a strong basis, they are a wonderful aid.
I would advise following a more traditional path, with a good book, but I cannot think of any good reason to not take a break from the book to watch some related 3B1b video from time to time.
Do you have any recommendation on a book about proofs? Every time I delve into algorithms most CS books seem to make large jumps in logic when demonstrating proofs.
The best book about proof I've read is "How to prove it" by Daniel J. Velleman. A lot of set theory. But to understand it, you must do the exercises in the book.
And for the solution, checkout this blog:
https://www.inchmeal.io/htpi/index.html
I'd like more math professors to make recordings of their courses. E.g. here is a pure gem [1]. This is a recording of a real analysis course from Rudin's principles of mathematical analysis (aka Baby Rudin aka the architect of many beginning mathematics students' misery). Anyone that's worked through this book knows the world of pain you're about to enter. I'd love if there was a similar recording for say, Charles Pugh's real mathematical analysis. I mean he's still alive, dunno if he still lectures from his book, but why not record one of those? It would be game changing for those who self-study.
Here's another request: Terrence Tao does a recording of a measure theory course.
Personally I find the Khan Academy, 3B1brown stuff to be a waste of time. It's too elementary and cutesy. That is not anything like what I've experienced real mathematics to be like. It is a continuous painful struggle.
> Personally I find the Khan Academy, 3B1brown stuff to be a waste of time. It's too elementary and cutesy. That is not anything like what I've experienced real mathematics to be like. It is a continuous painful struggle.
Sorry, but what?
You'd be hard pressed to find any single person, in the past 10 years, that has ushered more people into STEM by providing a solid fundamental education, than Salman Khan.
I understand that you're wanting more rigorous math, on a higher-level, but still - to blow off KA, 3B1, etc. as a waste of time sounds more ignorant than anything. Of the millions and millions that have seen these those guys, I'm sure a certain % have gotten their "Aha!" moments there, which have helped them tackle on much more comprehensive topics / fields.
edit: Just to clarify, I do not think these vids are any replacement for a solid book, just in the way I do not think lectures or tutorials are a replacement. But they can be fantastic aids for your learning.
Interactive websites are IMO vastly more efficient at conveying information, compared to static pages (like books, for example).
The value in good mathematical books is mostly the exercises. Like Spivak, Rudin and Pugh. Struggling through those is vastly more valuable than any video or hand holding. I mean most of those problems would take hours or even days to solve. I suppose Khan videos (or at least the ones I’ve watched) are useful but it’s no substitute for active thought. What I like about videos from a real course is that you can “see” how a real mathematician thinks about attacking a problem.
Err, uh, where do you get the idea that Khan academy "has ushered more people into STEM than any single person"? Seriously. I don't even know how you would measure something that.
Ok, so it's a very bold claim from my side - but he's done a phenomenal job making fundamental math (and science) lectures accessible, and more so, he does have a teaching style which works for a lot of people. It has been much faster paced, and more concentrated.
As for pure numbers, his videos have a total 1,806,946,640 views and 6M subscribes.
(Sure, there are scientists and inventors that have inspired magnitudes more people to enter STEM over the years)
It's probably way easier to decide on a subject you want to learn more about, and find the most approachable textbook / lecture notes on e.g. StackExchange. It doesn't make sense to choose courses (or even subjects to learn) based on whether video lectures are available.
I don't maintain a collection of these, but I can probably recommend
* Stephen Abbott, Understanding Analysis, for mathematical analysis and a little bit of real analysis
* David Williams, Probability with Martingales, for advanced probability
* Dym and McKean, Fourier Series and Integrals
Alternatively if you have a more cursory interest, you can probably check out Metacademy [1] which provides roadmaps to learning a variety of math (and ML) concepts.
Although I am a bit interested in number theory, linear algebra and a bit of calculus and its applications, I still haven't fixed my mind on specific fields. Now from time to time I watch videos on popular math YouTube channels so I can see what I like most.
I heard this from a senior colleague of mine that he's been working through the book "How to prove it: A Structured Approach"[1] that show how to prove things in mathematics, and has quite good exercises.
Perhaps a hands-on approach such as solving the exercises while working through this book will prove beneficial and is complimentary to watching, say, KA or 3B1B.
I think that book is better once you're already fairly comfortable with proofs. I tried to read it early on in my mathematics studies and couldn't understand it. Until you've done a number of proofs and start to see commonalities between them (e.g. proof by contradiction, induction, etc.) you don't have the mental scaffolding to learn more systematic approaches to proofs. I realize the book attempts to teach these techniques, but in my opinion it is hard to motivate these until someone has actually needed to use them to solve problems they're interested in.
Just found at least two books on topics of my interest and there are several more that I think I will read in the future. This is a good resource. Thank you!
As an alternative to getting a textbook, I would strongly recommend Krista King's maths courses. They're available through Udemy or her personal website [1][2].
Each section has a 1) written explanation, 2) video explanation with an exercise walkthrough and a 3) set of exercises. Each exercise also has a well-written walkthrough in case you can't solve them.
I've been working through them part-time for 4 months now, and it's honestly the first time that maths has clicked for me (I've found that most books tend to skip concepts which leave me lost).
I suggest spending less time with online lectures/videos etc. and more time with actual Books. The online stuff should be an adjunct to your study of books. Also it is best if you look at books dealing with a survey of all the relevant topics required for Science/Engineering before delving into specialized ones. To that end, i can recommend the following;
Mathematical Techniques: An Introduction for the Engineering, Physical, and Mathematical Sciences by Jordan and Smith - This book is really accessible.
Mathematical Methods for Physics and Engineering: A Comprehensive Guide by Riley, Hobson, Bence - A more advanced coverage of topics than the one above.
Mathematics for Physicists: Introductory Concepts and Methods by Altland and Delft - The coverage is excellent but somewhat challenging to read.
Has anyone here used it and could comment on its efficacy for people with say, good high school + some college courses knowledge of maths looking to grow? I've seen more than enough adverts for it on youtube but would love to hear thoughts from someone not being paid to hawk it!
Slight tangent given it's not all about "courses" per se, but my all-time favorite math-related website -- https://betterexplained.com -- is remarkably good at aiding in the development of intuition and deep understanding. Enjoy!
My own way to expand my knowledge was by going through textbooks and with a heavy support of Math StackExchange.
Maths is a very written-oriented domain, but still if you are more a video-oriented learner you can probably find a lot on Youtube, and a lot of great researchers have videos there.
I am not sure what your "discrete math courses" were and if you have already taken these but, the Linear Algebra and Differential Equations courses on Kahn Academy are solid intros to the basics of those subjects.
Doing Khan Academy then 3Blue1Brown got me through many mental blocks when it came to math stuff. I used KA during university and it really helped me, but wish 3BlueBrown existed back then as it has even better visualizations - this was key for me to make math interesting and to see why certain levers exist in math and what happens when you pull them (how changing formula variable values changes outputs).
If you are in India, NPTEL courses are a possibility, but I've heard that it can be a hit or a miss depending on the lecturer. You get to write regular assignments, and a proctored exam for a certificate.
I’m working through the entirety of brilliant.org right now; some of its courses are specifically maths, and the skill level goes from introductory courses all the way up to maths-degree level topics.
My take: it's good but has some drawbacks. The biggest one for me is that the questions are mostly multiple choice. You don't have to show your work or reason through the problem, so it's easy to mentally cheat and say "eh I think it's this one." This isn't so much a fault of Brilliant as a limitation of the smartphone/web technologies vs. pen and paper. I've been having some success with a blend of Brilliant, books, and application (using what I learn to read some physics). I don't think using it on its own will probably be enough though if you want to be able to apply or know the topics deeply.
As much as I really hate to say it, youtube seems to have the best collection of recorded math lectures I can find anywhere. OCW has quite a few but only the ones from MIT. One of my professors only uploaded to youtube (and I think he was doing this without the administration knowing, they were generally hostile to recorded lectures and the official policy was that recordings of lectures should never be shared to non students for some reason.)
Besides that good math books are usually very cheap. I would highly recommend those over a website.
I highly recommend http://schoolyourself.org/ over just about anything else (Khan Academy, etc.) It's free, and has a great interactive user experience. Algebra through Calculus.
I'm in the same boat as you and I just started taking classes at my local community college/university. Options are affordable and lots of online offerings.
My advice would be to get a book on any topic which interests you, read through it, and do a significant number of the exercises. You might try Epp's Discrete Mathematics, Hefferon's Linear Algebra, Colley's Vector Calculus, Dudley's Elementary Number Theory, Spivak or Apostol's Calculus (these go far beyond ordinary freshman calculus), Pinter's A Book of Abstract Algebra, among many others. Some of these books are expensive to buy new, but just buy older editions.
Resources like Khan Academy and 3blue1brown are also fantastic, and I have shared some of these with my students. I'd recommend using these as a supplement; if you rely on them solely then you'll develop vague intuition but not much else.
Also, with the pandemic, there are a huge number of traditional university courses that have moved online, and you could probably enroll in one for not too much money. Check the RateMyProfessor reviews -- you want a mix of positive reviews claiming the prof inspired them, and negative reviews complaining that work was expected. I have a RateMyProfessor review which complains bitterly that "homework is graded for accuracy and not completion". :)