Ask HN: How can I learn math?
When I was younger, I had a lot of problems. Ended up in a reform school, and they weren't very focused on education.
Fast forward ten years. I've done a lot of self learning... mostly statistics, but there are huge gaps in my knowledge.
Too often in my career, I've decided on a project or course of study based on my math skills rather than my passions.
The problem is, I don't even know where or how to begin. I don't know what I don't know. And whenever I try to look up a specific concept (say, the Wikipedia page on Calculus) there's so much more that I don't know that the gulf of execution is so great I end up just giving up.
Does anyone know of a good online resource to sort of start at the beginning and work my way up?
(For example, I used to also hate learning foreign languages, but discovered Duolingo and have been working on learning German every since. And it's actually kind of fun!)
28 comments
[ 2.9 ms ] story [ 75.1 ms ] threadKhan Academy (https://www.khanacademy.org/) has great resources for all levels of maths up until the first year of university - easily the friendliest and most comprehensive set of classes and topics for maths until that level
MIT Open Courseware (http://ocw.mit.edu/courses/find-by-topic/#cat=mathematics) has many courses that you can pick from and start to learn from. For these it doesn't hurt to see what textbooks they're using (if any) and purchasing them and going through the problem sets yourself.
The great thing about maths, is that until you get to the very high levels, many problems can be checked against pre-made answers.
Hope this helps!
To add to this, wolframalpha has proven to be a huge help to me. It has solved every problem I've fed it so far, the pro version even shows you the intermediate steps.
After this, I suggest learning some discrete math and proof techniques. The book How To Prove It is great. It will teach you logic, set theory, how to write proofs, and how to invent proofs. Learning this first will help you actually understand calculus when you study it next.
For calculus, MIT's OCW course is really good. Pick up a standard book like Stewart, do a lot of problems, and try to understand the proofs of all the theorems. Or if you'd really like a challenge and some more theory, pick up Spivak's book.
I like Mooculus, developed at Ohio State http://mooculus.osu.edu
To figure that out, the most helpful thing I've found is looking at example 4-year plans at colleges (and, if they're available, even for some high schools when it comes to the fundamentals of a subject I never took), and, for online any given online course, seeing if there are any recommended prerequisites or co-requisites.
As an afterthought to all that, my favorite tutorial sites are like, HyperPhysics for physics, MIT open courseware for CS topics, HowStuffWorks for general tech-y knowledge (say, if I wanted to learn how a capacitor or a web server worked). If I'm going to google for good tutorials, I usually include something like "tutorial", "introduction", "primer", "layman's guide", or "cheat sheet". I find that even if I'm looking for an in-depth learning experience, the tutorials that are written to be simple will do the best job of emphasizing what's important, and laying out the way that somebody who "knows how to do it" will approach a problem.
and this: http://www.amazon.com/dp/0486409163/?tag=stackoverfl08-20
Also, you might want to get Richard Courant's "What is mathematics?" and a book (I'm reading multiple as I'm in your same spot, didn't have much mathematics during high-school because I thought I didn't possess the acumen, then I realized I really liked the subject) on proofs.
Gelfand's books are very very very good, trust me on this one, they build on the fundamentals. The books are not short of flaws though, namely the writing is informal, the author assumes some preexisting knowledge (that's why they are often not used as class books but as supplementary notes) and do not offer many exercises. But if you get the whole bunch you'll have covered the high-school curriculum (and more). The AMA olympiad books are good reads, same with those "Art of Problem Solving" books. But personally, I'm not starting these until I've gained enough confidence, I still can't solve elaborate problems or mathematical olympiad kind of questions (but I'm getting smarter).
[1] http://betterexplained.com/articles/category/math/
You need two components to properly learning math: a) theory b) practice
# THEORY
Sequence: 1) Algebra 2) Geometry 3) Trigonometry 4) Calculus (single, multivar calculus)
Here's links for (1)––(3) http://www.mathsisfun.com/algebra/ http://www.mathsisfun.com/algebra/index-2.html http://www.mathsisfun.com/geometry/ http://www.mathsisfun.com/algebra/trigonometry.html
For calculus, I don't know anymore productive way than taking classes at a local Junior College. You need to place into these classes since you haven't taken math classes in a few years. Try to place into a compact summer class. My local college lets me place into Calc 1 & 2 (Honors) as the highest class to place into.
The website (mathisfun) is a great resource. It looks really kiddy, but concept are explained properly. If you can get over the visual aesthetic, you can relearn mathematical ideas quickly.
# PRACTICE
Buy a book. Preferably a book that has a boat load of problems for you to run through.
Build confidence through exercises.
I recommend you get a proper book, which is going to be more complete, will start at the basics, and then build on itself. Anyways, I recommend you learn algebra, geometry, trigonometry, and calculus, in that order. You'll also want to learn linear algebra, but you should be able to understand it after basic algebra. I can't recommend a book for algebra or trig, since I took them so long ago. Calculus by Stewart is a popular text book, is accessible, covers the complete basics, and has old editions cheaply available. (I bought mine for $5. Older editions of textbooks are dirt cheap, and have almost the same content as newer editions.) Plenty of people don't like it, and there might be better calculus text books, so I'm not saying it's the best. Strang has written several books on linear algebra, they are well written, but not necessarily thorough. Once you have a textbook on the topic you're interested in, use it to accompany Khan Academy. Math builds on itself, so you'll constantly be referring to previous stuff that you learned, and this is significantly easier with a text book. Mark it up, highlight every definition and theorem in it, and never through it away. Check out your local library, and see what books work for you, then buy them. If you have a college/university near by, their library will have the books that their math department uses. Note that they might not be on the shelves, you'll have to ask the front desk for it, and you can only borrow it for several hours.
Once you have a decent understanding of calculus, read a proper book on math thought/proof writing. The class I took on this changed my life. All upper level math books are extremely structured, and this will teach you how it is done, as well as how to structure a proof, and set notation. I read Mathematical Proofs by Chartrand, but there are others. Once you have done this, you can easily teach yourself any math topic and have the ability to understand any math paper. You can now learn real analysis and/or abstract algebra (I recommend Pinter for abstract algebra).
TLDR; Learn math from proper text books, use online resources to help you get through them. Learn algebra, geometry, trig, calculus in that order.
When finished, buy the next grade ups textbook.
Learning math needs paper, it just does, don't try to do it online, it'll take twice as long and you'll learn half as much.
Be active not passive. Always learn with scribble paper ready.
This book is my personal favorite.
http://www.amazon.com/Calculus-Practical-Man-J-Thompson/dp/1...
You can look up "The Cartoon Guides..." For example, they do have a "Cartoon Guide to Statistics." I used to own it. I can tell you that the first chapter or two covers what my college intro to statistics covered. I was inducted into Mu Alpha Theta, a math honor society, when I was 16 and I tutored math. I am good at explaining it and have a bit of a background, though I am a math slacker for HN. So, you know, we need more context to figure out what "beginning" you are looking for.
I will also say that I was in my thirties before I understood that the formulas I memorized my way through in high school had actual real world applications and so on. I was clear my oldest son could not just memorize his way through formulas. People who can do that are inclined to be math majors. People who cannot probably should not take much advice on learning math from math majors (I mean, unless they are experienced teachers as well who know how to reach folks who aren't so good at math -- Colin Wright's juggling video is very approachable, but many math majors seriously suck at explaining math to people who aren't also just inherently good at math). They are two different kinds of minds.
Best of luck.
The best and worst thing today is that you have a huge number of resources. So much that you can either learn it all, or none. Most will go with none.
Everyone will recommend their favorite books. I'll do the same:
- Piskunov's "Differential and Integral Calculus".
- Demidovich's "Problems in Mathematical Analysis".
Piskunov for a course (concept, example. concept, example) and exercises. Demidovich for a quick review and about 3000 exercises.
I personally prefer the Soviet style for Maths and Physics. They're sharp, read your mind, and are ADHD free.
Despite already having my degree, I've felt for a long time that I've wanted to REALLY learn this stuff, at least to a point where I can read through Introduction to Algorithms and "get it".
My base level of knowledge is probably the start of Algebra 1, so I've been going through Khan Academy to build myself up. I'm halfway through Algebra 1 and I've already come across a ton of stuff that I barely ever covered in my GCSE's. I can't vouch for Khan Academy enough. It's been a far better teacher for me than any I've had.
I've given myself around two years to complete the following in my own time:
* Algebra 1 and 2
* Calculus
* A read through of Knuth's Concrete Mathematics
* A read through of Introduction to Algorithms and TAOCP.
I'm part-way through the first one, and I'm hoping that if I stay consistent (an hour of Khan Academy a day, and maybe a bit more on the weekends) I'll be able to work my way through this list.