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Ask HN: Is there a website specialized in mathematics courses?

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Re: Ask HN: Is there a website specialized in mathematics courses?

#51

Post by jeffreyrogers »

smukherjee19 wrote: post #27

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. [1]: https://www.amazon.com/How-Prove-Structured-Dan…

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.

Re: Ask HN: Is there a website specialized in mathematics courses?

#52

Post by laichzeit0 »

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.

[1] https://www.youtube.com/watch?v=ab41LEw9oiI&list=PLun8-Z_lTk...

Re: Ask HN: Is there a website specialized in mathematics courses?

#54

Post by impendia »

wcarss wrote: post #50
impendia wrote: post #38

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…

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 thi…

Very good question.

This is difficult. One way to do this is to participate on Math Stack Exchange:

https://math.stackexchange.com/

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:

https://www.math.wisc.edu/tutors

An hour every two weeks would go a long way.

Re: Ask HN: Is there a website specialized in mathematics courses?

#55

Post by artagnon »

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.

https://artagnon.com

Re: Ask HN: Is there a website specialized in mathematics courses?

#56

Post by TrackerFF »

laichzeit0 wrote: post #52

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 P…

> 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).

Re: Ask HN: Is there a website specialized in mathematics courses?

#58

Post by rcollyer »

wcarss wrote: post #50
impendia wrote: post #38

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…

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 thi…

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.

Re: Ask HN: Is there a website specialized in mathematics courses?

#59

Post by jimhefferon »

wcarss wrote: post #50
impendia wrote: post #38

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…

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 thi…

> 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

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