Ask HN: How can I learn math?
22 コメント
https://brilliant.org is another good source.
What level of maths are you exactly looking at?
Khan 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!
Khan 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!
> The great thing about maths, is that until you get to the very high levels, many problems can be checked against pre-made answers.
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.
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.
Will second the pro version thought - IIRC, the app is like 2.99, but a bargain at that.
Have you mastered algebra and trig? If not, do it. Khan Academy is great for this. Watch the videos and do lots and lots of problems. In learning math, there is no substitute for working through lots of problems.
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.
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'll be a third person to recommend Khan Academy.
There are great courses on http://www.coursera.org and http://www.edx.org
I like Mooculus, developed at Ohio State http://mooculus.osu.edu
I like Mooculus, developed at Ohio State http://mooculus.osu.edu
Ok, everybody is recommending different content sites, so you've got places to go to find specifics. That's good. But you also need a way to figure out what you already know, what you want to know, and which order is best to learn stuff in so you're not confused. For instance, you'd better have a decent understanding of trigonometry, geometry, and pre-calculus before you try tackling calculus. The hardest part about self-learning isn't trying to find tutorials for what you want to learn, it's figuring out how the specific topic you're working on can be contextualized in terms of other topics in the same subject area.
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.
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.
OH I ALMOST FORGOT - Paul's Online Math Notes!!! That site got me through high school maths!
I would highly recommend the book 1089 and All That. It won't teach you much math, but it is an excellent book to read while getting started. Measurement is also a good book that starts from scratch.
Buy those:
http://gcpm.rutgers.edu/books.html
and this: http://www.amazon.com/dp/0486409163/?tag=stackoverfl08-20
and this: http://www.amazon.com/dp/0486409163/?tag=stackoverfl08-20
Unable to open the amazon link, could you please repost or mention the name of the book ?
Mathematics: Its Content, Methods and Meaning (Dover Books on Mathematics)
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).
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).
Mathematics: Its Content, Methods and Meaning (Dover Books on Mathematics)
Do not get discouraged. Wikipedia is an encyclopedia! Math education is not the goal of an encyclopedia. Other sources in this thread will help.
To help build your intuition when learning math you might find these articles helpful [1] as supplemental material.
[1] http://betterexplained.com/articles/category/math/
[1] http://betterexplained.com/articles/category/math/
Don't listen to anyone here that even mentions Khan Academy.
It's good if you don't mind wasting a lot of time (I'm assuming you're a remedial learner).
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.
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.
Applied math major here. I strongly recommend you don't use websites to teach yourself math, with the possible exception of Khan Academy or OCW. Most websites or YouTube are only good if you already somewhat know the math topic, and you just want a different way of explaining it to help reinforce the topic. Most websites feel like tutoring: only good for helping as a secondary source. Wikipedia is completely useless for me unless I already know the topic, in which case it is a decent reference. I've tried to teach myself math using online resources, and failed. (Note that I have taught myself C, C++, C#, Java, and Python using online resources, and I know how to use Google, but math is a lot more formal.)
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.
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.
Go to the dead paper store and buy the textbook for the grade level you want to start with. This way you get a complete, comprehensive guide with table of contents that you can work through. Read, and do the sample questions.
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.
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.
I can relate to the same problem and often I am stuck when reading about algorithms explained through math formulas. I would love to see some kind of dependency tree for math knowledge which clearly shows all the requisites for understanding a specific topic.
Maybe it is a nice problem to solve for empathic math freak enterpreneurs which can come up with new, structured teaching methods.
I would recommend starting with a calculus book. If you aren't comfortable enough with the trig side of things, do a refresher on OCW or Khan Academy.
This book is my personal favorite.
http://www.amazon.com/Calculus-Practical-Man-J-Thompson/dp/1...
This book is my personal favorite.
http://www.amazon.com/Calculus-Practical-Man-J-Thompson/dp/1...
Start here: http://us.metamath.org/mpegif/mmset.html
We kind of don't know enough about where you are to be able to help you. I homeschooled my gifted-learning disabled sons and I strongly disagree with some of the things suggested here.
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.
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.
A book called Mathematicians Delight will be very helpful. Easy readying. A classic.
Nothing to be embarrassed about. I graduated and I'm still working on my Maths and Physics daily. I have a folder in my computer named "Core Knowledge" because I noticed that whenever I struggled with a topic, it wasn't really the topic I was struggling with, it was my core knowledge that was weak. I'm strengthening the core so that I'll only have to deal with the topics.
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.
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.
Khan Academy is great. I wish they had some kind of placement thing to evaluate your skill in different areas of math, but you can always just start with something you recognize and work your way up from there.
I hated math before and I thought I was gonna hate it in my whole life. Not until I had to study GMAT and it changed my life forever. I watched the GMAT problem solving explanation on Khan Academy. I think it was pretty simple math and fun to learn. easy to understand for math haters. It was pretty awesome. To me, math is the new door to the new world. I started reading books about math and related field like Physics. (Thinking in numbers by Denial Tammet, How Not to Be Wrong by Jordan Ellenberg. Cosmic Numbers: The Numbers That Define Our Universe by James D. Stein, Richard Feynman and etc. I think you get this, stay curious, stay open to learn. Hope this help and good luck! :)
I have a degree in Computer Science, but I am terrible with Maths. It's surprising how far someone can go without ever doing calculus if they're able to fit enough answers in their head.
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.
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.
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!)