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Ask HN: How to self-learn math?

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Re: Ask HN: How to self-learn math?

#21
Whenever I tried to self-learn anything, it was a very bad idea. Some parts of the subject I enjoyed and other parts I hated. I tend to make my own conclusions, which parts are useful and wich are a waste of time (so I tend to skip them). I tend to filter out the material this way, in order to make learning less painful and more fun.

My conclusions (what is useful and what isn't) were always wrong and I ended up not learning anything properly, not getting a proper understanding of anything.

Please, if there is still any such option for you in your country, always choose a proper school education instead of self-learning. It is really great, when there is some leader with a proper understanding of the subject (a teacher) and others, who are having "the pain" with you (classmates), so you can see you are not "suffering" alone, and you don't start making your own conclusions (since you would see, that others are taking seriously what you wanted to call a waste of time). Classmates also help each other during the learning process.

So personally, I think a person gives up self-learning as soon as it becomes too painful / boring. The best way to overcome it is to see other people around you going through the same process, or to see somebody who you admire, who has already gone through the same process (it could be your teacher, your parent, your role model etc.). You could call that "the motivation".

Re: Ask HN: How to self-learn math?

#22
I would select books based on your interest. I find the Dover publications good because they are both cheap and slightly older, this means they are less focused on undergraduate monetisation (version hopping, not supply answers to problems, glossy print), and more focused on proofs and algorithms. You can see them here http://store.doverpublications.com/by-subject-mathematics.ht.... I particularly liked Probability: A concise course, and Number Theory by George Andrews.

Amazon used to have a great number of graduate preparation book lists which always included books such as Rudin's Principles of Mathematics Analysis, and Halmos' Finite Dimensional Vector Spaces. These classic maths books are brilliant but usually easier to understand if you already have some experience with the material.

Final advice is to find a study partner as it can be hard to track how you are going and keep motivated, especially without the instant feedback loop you get with programming.

Re: Ask HN: How to self-learn math?

#23
You want to acquire and shoot for the so-called mathematical maturity. More precisely: to become an autonomous problem-solver and have the know-how to solve (non-)trivial proofs. Typically this means bridging the gap between computationally based maths which one is exposed to in pre-school to high-school years and sometimes in the first year of college/uni, and proof-based maths which involves and demands a good command of sets and operations on sets, quantifiers (universal, existential), logical operators (not, and, or, material conditional, biconditional), and proof methods (direct, indirect a.k.a reductio ad absurdum, induction, pigeonhole principle, etc.)

A good series of books aimed for pre-school and high-school students to accomplish just that is The Art of Problem Solving. Google it.

Re: Ask HN: How to self-learn math?

#24
I agree with other comments that "learn maths" is too broad. You can take a university degree in maths and still be just at the beginning of "learning maths." I recommend refining your goal somehow: perhaps to learn math related to certain applications that you're interested in, or learn math in a certain area (e.g. high-school algebra, geometry, probability, discrete math, graph theory, calculus, pure math, abstract algebra, topology, etc).

If you have not mastered high-school algebra and other pre-calculus subjects, you should start there; most other maths subjects will assume that you know these things. Calculus takes up a lot of space in upper high-school and early university courses -- but if you're a developer there may be other subjects that are more immediately useful to you (e.g. discrete math, linear algebra).

I set out to "learn maths" (that's verbatim what it says on my personal Kanban board). In the end I took some university classes. For me they provided the structure and teachers to help me learn. Also, there is a difference between having an idea about what some math-thing is, and being able to pass an 3 hour closed-book exam in that topic.

I agree that Khan Academy is a good learning resource that will provide structure to your learning:

https://www.khanacademy.org/

Purplemath is another good resource:

http://www.purplemath.com/

YouTube is full of videos of people running through problems on any conceivable topic. Definitely search there for help.

Once you've worked your way through the high school prerequisites, I'd recommend Linear Algebra as a good next course. It has many practical applications, and is also an entry point towards pure math subjects like Abstract Algebra. Also, you don't need to know any calculus to study linear algebra. I like Gilbert Strang's OCW course:

https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra...

Finally, mathematics is HUGE. The following will give you a bit of an idea:

The Map of Mathematics https://www.youtube.com/watch?v=OmJ-4B-mS-Y

Re: Ask HN: How to self-learn math?

#26
Method-wise it could be helpful to get a (lightweight) computer algebra software and learn how to use it and how to explore knowledge using it. One thing you won't have when you're out on your own is a method to just try out stuff and verify that it is correct, or to get better visualizations quickly. Often you will get stuck with something and need a different angle (which teachers or other students could normally provide). Then you can just open the software and play with it.

One place to do that for free on a basic level would be Wolfram Alpha: https://www.wolframalpha.com/examples/math/

Edit: (I mean this in addition to the learning resources like books and videos)

Re: Ask HN: How to self-learn math?

#27
post #19

Ok, I'll take a crack at this: Up to high-school level: 1. Precalculus: Precalculus: A Prelude to Calculus - Axler 2. Calculus: The Calculus Tutoring Book - Ash. College: 3. Preparation for Collegel-level maths: 3a. General prep for high level maths: How to Study as a Mathematics Major - Alcock 3b. Proof writing: How to Prove It - A Structured Approach - Velleman OR Book of Proof (2nd ed) - Hammack (it's free!) 4. Ma…

Two quick things I can recommend without hesitation, which focus on an intuitive understanding of concepts:

1. Essence of Linear Algebra mini-series - https://m.youtube.com/watch?v=kjBOesZCoqc

2. Better Explained website - https://betterexplained.com

YouTube has a lot of high quality math content, it definitely helped through university. It's also worth mentioning the Stanford U courses.

The main takeaway I have for you is learn the concepts intuitively first, then spend the time to play around with them on paper until they sink in. Some things will be easy, some will be frustrating, much like programming you will walk away from a frustrating problem and have an epiphany while doing something completely different.

All the best and have fun!

Re: Ask HN: How to self-learn math?

#28

Method-wise it could be helpful to get a (lightweight) computer algebra software and learn how to use it and how to explore knowledge using it. One thing you won't have when you're out on your own is a method to just try out stuff and verify that it is correct, or to get better visualizations quickly. Often you will get stuck with something and need a different angle (which teachers or other students could normally p…

This is good advice, and Wolfram Alpha is magical!

Re: Ask HN: How to self-learn math?

#29
I am not a maths major, however as I currently self-studying Mathematics, so I hope this would come as a good reference points for you.

I think we both should prepare for a long journey, cause it is the nature of maths.

I prefer formal and classic textbooks/notes as I think they are the best resources. Mathematics has been around for a long long time, keeping things up to date isn't really what should most concerns you.

[0] : https://www.quantstart.com/articles/How-to-Learn-Advanced-Ma... This article aims at kick-starting a career in quant, but the bullet points are really similar to any undergraduate program.

[1] : Schaum's series Really good textbooks on basic maths, helped me a lot on those maths modules during my study.

[2] : Any Massive Open Online Course of your choices. I am currently using MIT OCW. They are basically an Undergrad Course minus interaction with lecturer. You should ask some of your maths friends to help you out. Good, intuitively explanation in person helps a lot.

[3] : And last but not least, have fun while doing it. You can participate in maths competition, watch Youtube videos( 3Blue1brown / Numberphile) Read Magazines and Journals too, admires the Apollonian aesthetic of Mathematics.

Maths is one of the few subjects where nature > nurture, I think ( and observed). But take heart.

Re: Ask HN: How to self-learn math?

#30
post #19

Ok, I'll take a crack at this: Up to high-school level: 1. Precalculus: Precalculus: A Prelude to Calculus - Axler 2. Calculus: The Calculus Tutoring Book - Ash. College: 3. Preparation for Collegel-level maths: 3a. General prep for high level maths: How to Study as a Mathematics Major - Alcock 3b. Proof writing: How to Prove It - A Structured Approach - Velleman OR Book of Proof (2nd ed) - Hammack (it's free!) 4. Ma…

To the above I would add:

How to Think Like a Mathematician - Kevin Houston (an excellent book to read before starting)

How to Read and Do Proofs - Solow

The Keys to Advanced Mathematics: Recurrent Themes in Abstract Reasoning - Solow

Calculus - Spivak (Actually a Real Analysis book, not a Calculus book, see e.g. https://math.stackexchange.com/questions/1811325/spivaks-cal... )

Linear Algebra Done Right - Axler (Intended for a second course in Linear Algebra, but I found it helpful during my first course.)

And for something from left-field:

Visual Group Theory - Carter http://web.bentley.edu/empl/c/ncarter/vgt/

There are many many many books on every mathematics topic under the sun. Finding books that speak to you is important. I have had mixed success buying books upon other people's recommendation. You would be best to get access to a library.

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