I recently met up with an old friend, and asked how high up he's gone with mathematics. I said "So, what, like algebra?" He said, "Yeah." I can see that he is genuinely interested in mathematics, but he's in a situation where he can't feasibly get a college education like myself, and I want to point him to a resource on building up his mathematical wit without just saying "Just wikipedia it." There is a build up of concepts that tie together with structure, and I just wonder what this community would recommend as an online resource for building a strong foundation in being able to touch higher level mathematics, say, at a senior undergraduate level. A framework of topic precedence that can be used as a guide. I wanted to lend him my "Art of Problem Solving" book series, which I think does a good job, but I think he's into something more open source and digital. Also, these books were kinda expensive.
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1well for one ,you could get him into the stackexchange community . Also see if he is able to obtain scholarships in mathematics institutes . – The Integrator Apr 03 '18 at 05:16
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@pranavB23, well, that's not an option, i.e., "mathematics institutes." He's a father, and it's clear to me that the university doesn't appeal to his lifestyle. – dr_bubble_pops Apr 03 '18 at 05:18
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This is, as I see it, asking for personal advice (although it's not for you but your friend), which is off-topic on this site. – Arthur Apr 03 '18 at 05:22
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1@Arthur, in other words, are you saying it can't be objectively evaluated. – dr_bubble_pops Apr 03 '18 at 05:25
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Well, yes. Also, in order to be able to give a good answer, specifically to your friend, and not just general advice, we would need to know a creepy amount of information about your friend, his current life situation, and what he has been up to mathematically. No one wants that. – Arthur Apr 03 '18 at 05:27
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3Wikipedia is terrible for learning math by the way. – Matt Samuel Apr 03 '18 at 05:34
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@Arthur: If the question is interpreted as asking for online resources for a generic person for whom joining a university is infeasible (for whatever reason) then it's a perfectly fine question. dr_bubble_pops: You could consider editing the question to make it more broadly applicable in this way, to head off such objections. – Apr 03 '18 at 05:36
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1@dr_bubble_pops Wikipedia is great as a reference. Not so much for learning in the first place. It's too fractured and the language is quite advanced most of the time. – Arthur Apr 03 '18 at 05:37
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Online course material, such as those on MIT OCW might be a good follow-up for topics your friend becomes interested in after seeing them on 3b1b/Mathologer. There are even some video lectures on OCW, if your friend is into those. – Theoretical Economist Apr 03 '18 at 05:53
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2DoverPublications.com, known in the paper era as Dover Press, was, and still is, a great source for very cheap re-prints of math books, including many old timeless classics. – DanielWainfleet Apr 03 '18 at 08:32
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@dr_bubble_pops: You do know that the tag has existed pretty much since the conception of the site, right? And that the definition of acceptable has changed over the years, right? – Asaf Karagila Apr 03 '18 at 08:39
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@Asaf: How is any new user supposed to know that?? – Apr 03 '18 at 09:07
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@Rahul: By looking around. My comment is a reply to the user posting a screenshot of the tag's card showing there are 8.1k questions tagged as [soft-question], as if that justifies their question. – Asaf Karagila Apr 03 '18 at 09:08
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Wow, the question that "can't be objectively evaluated as correct or incorrect" is primarily opinion-based. Who'd'a thought? – dr_bubble_pops Apr 03 '18 at 18:52
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@dr_bubble_pops: Your attitude reveals a lot here. You know? In your single day of being a member, you apparently know all the ins and outs of the site. That's very impressive. – Asaf Karagila Apr 03 '18 at 23:38
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@AsafKaragila, explain. – dr_bubble_pops Apr 04 '18 at 05:11
3 Answers
This is not a complete answer by any means. Just one of many ideas.
There are a number of quality YouTube channels out there. 3Blue1Brown is one that immediately springs to mind. His linear algebra playlist in particular is rock solid and a phenomenal supplement to a textbook. I think he has one for calculus as well, plus individual primers on a myriad of topics from Fourier analysis to blockchain technology. Many of them help whet the appetite for rather advanced topics while still being surprisingly accessible to someone with little formal mathematics education. Others, like KhanAcademy, have a wider selection, especially for earlier mathematics. Search around!
Some reasons I recommend YouTube channels are as follows:
I have a tendency to get bogged down by notation, and unfortunately this notation is rather unavoidable in a pure-text format. At least for me, seeing a lot of ink on a page can be intimidating, and often it's not even that difficult of a concept. Being able to watch computer-generated animations or someone drawing diagrams or even just listening to an expert's informal stream-of-consciousness can do wonders cutting through the notational barrier.
There are a number of subjects where it's absolutely essential to have lots of visual representations of the mathematics you're seeing; imagine trying to go through vector calculus without seeing many pictures. This is just one instance where having a professor in an actual classroom can be really useful, but even a professor can't animate drawings on a chalkboard, and some have limited drawing skills.
It offers yet another perspective on the material being learned. One should never confine themselves to a single source when learning new concepts. Often, there are vastly different approaches to presenting a concept, and seeing as many as possible affords richer understanding. Plus, if you get stuck in one source, another might help clear things up.
Coupled with Math.SE as a resource, one can link directly to the very moment one gets stuck or a question arises.
Watching YouTube videos, of course, comes with a few caveats. First, they should be a supplement, not a substitute, for actual textbooks. And second, it's easy to become too passive. Don't just watch. Work problems. Put pen to paper. Doing exercises yourself is the only way you're going to internalize what you're trying to learn. You may think you've mastered something, but all-too-often, holes in your understanding aren't going to become apparent until you're actively doing the mathematics.
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Yeah 3b1b has introductory lectures only for calculus and linear algebra, wish he had more of these series – King Tut Apr 03 '18 at 05:30
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Also, khan academy and brilliant.org are places I've heard great things about. @KingTut As far as I can tell, he wishes that himself too. I think he has mentioned that he has both group theory and probability in the pipeline somewhere. There is a tremendous amount of work behind such a series, though. – Arthur Apr 03 '18 at 05:30
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@Arthur brilliant is good for problems, but to me it lacks structured course, so beginner might find himself confused. There are structured courses but they are paid, i think. – King Tut Apr 03 '18 at 05:40
Maybe you can show him some good sites for it, for example 3Blue1Brown or Mathologer,these channels mainly focus on the intuition and understanding of mathematical concepts . Khan Academy is also a good site I'd recommend as this also focuses on the way one can solve problems. Books and other resources are also a good place to start. I'll leave some links to sites which are really good and can help improve mathematical understanding .(1)PBS Infinite Series (2) standupmaths (3) Numberphile.
Also, he can always opt for online courses in mathematics , also I'd recommend this Paul's Online Math Notes if he is interested in some good notes.
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