Ask HN: How can I discover math?
So I've been trying to teach myself K-12 math using www.khanacademy.org now after reading http://www.maa.org/devlin/LockhartsLament.pdf I feel like I should be discovering math, playing with numbers asking more questions. The problem is that from up to this point in my life all I have been taught was to memorize a formula, plug in some numbers, get an answer. I want to learn math well, I want to enjoy the beauty of it, but I don't know where to begin that journey or how to start.
23 comments
Get the book "Thinking Mathematically" (http://www.amazon.com/Thinking-Mathematically-J-Mason/dp/020... non affiliate link) which exposes the process of how mathematicians think, using simple problems. You can then adapt the process to your preferred level of problems. Also learn proof technique (Velleman's How to Prove it" is a good book for this) and doesn't make any assumptions of mathematical knowledge.
Check out this previous post: http://news.ycombinator.com/item?id=1449799
And also, I'd highly recommend art of problem solving. http://www.artofproblemsolving.com/
Honestly, the best way to discover math is the Hacker's way i.e. find problems to work on and then start working on them even in you don't know how initially. The very digging in process will lead to further discovery and further learning. Now, be sure to check out art of problem solving forums, people ask questions there and there are very good detailed explanations available.
Be also sure to check out Street Fighter math (http://ocw.mit.edu/courses/mathematics/18-098-street-fightin...) and fermi questions (e.g: http://mathforum.org/workshops/sum96/interdisc/classicfermi....)
And remember to print out these questions and carry them with you along with tons of scrap paper. Sometimes, it helps just to read a problem and let it sink in over days...
And also, I'd highly recommend art of problem solving. http://www.artofproblemsolving.com/
Honestly, the best way to discover math is the Hacker's way i.e. find problems to work on and then start working on them even in you don't know how initially. The very digging in process will lead to further discovery and further learning. Now, be sure to check out art of problem solving forums, people ask questions there and there are very good detailed explanations available.
Be also sure to check out Street Fighter math (http://ocw.mit.edu/courses/mathematics/18-098-street-fightin...) and fermi questions (e.g: http://mathforum.org/workshops/sum96/interdisc/classicfermi....)
And remember to print out these questions and carry them with you along with tons of scrap paper. Sometimes, it helps just to read a problem and let it sink in over days...
Soviet textbooks and the Princeton Companion (recommended by others) are likely to be pretty brutal for you. I suggest problem-solving as a better way in.
Take a look, e.g., at Project Euler (http://www.projecteuler.net/); find some books aimed at K-12 students doing mathematical contests (there are lots; what's best depends on how much you already know) and online resources aimed at the same audience (e.g., http://amc.maa.org/, http://www.mathcomp.leeds.ac.uk/); there are other sources of not-too-routine mathematical problems online, such as http://nrich.maths.org/.
This pretty much guarantees that you'll be doing as well as passively absorbing (one of many problems with which is that it's easy to think you're absorbing when actually you're not). There's a very wide range of levels of difficulty. And it's likely to be fun, if mathematics really suits you at all.
Take a look, e.g., at Project Euler (http://www.projecteuler.net/); find some books aimed at K-12 students doing mathematical contests (there are lots; what's best depends on how much you already know) and online resources aimed at the same audience (e.g., http://amc.maa.org/, http://www.mathcomp.leeds.ac.uk/); there are other sources of not-too-routine mathematical problems online, such as http://nrich.maths.org/.
This pretty much guarantees that you'll be doing as well as passively absorbing (one of many problems with which is that it's easy to think you're absorbing when actually you're not). There's a very wide range of levels of difficulty. And it's likely to be fun, if mathematics really suits you at all.
"Discovering math", depending on what you mean, could be setting the bar pretty high. Memorizing formula is not what mathematics is about, but you're unlikely to get very far if you just "play with numbers". To learn mathematics and to think mathematically, you still need a structured environment: Lockhart's complaint is that modern (K-12) math teachers neither have that knowledge nor the ability to provide that structure.
If I were you, I'd go to where that knowledge does get taught: in undergraduate mathematics classes. That usually starts with some rigorous calculus (I second the recommendation for Spivak), real analysis (which might seem hopelessly unmotivated if you don't know/remember much calculus), linear algebra or abstract algebra, or discrete math. Depends on what you're interested.
Reading mathematics is hard. Google around for advice. Do lots of problems, which generally means finding proofs. This would very well fit the definition of "discovering mathematics".
If I were you, I'd go to where that knowledge does get taught: in undergraduate mathematics classes. That usually starts with some rigorous calculus (I second the recommendation for Spivak), real analysis (which might seem hopelessly unmotivated if you don't know/remember much calculus), linear algebra or abstract algebra, or discrete math. Depends on what you're interested.
Reading mathematics is hard. Google around for advice. Do lots of problems, which generally means finding proofs. This would very well fit the definition of "discovering mathematics".
You asked about beauty and history, with an idea on how to discover... personally, I started with "The Math Explorer: A Journey Through the Beauty of Mathematics": (http://www.amazon.com/gp/product/1591021375). That book got me hooked and from there I went on to pre-cal studies at university, eventually completing sequences in discrete structures, probability, up to and including calculus III, a level I never imagined I'd get to. That book was literally my starting point since I never took high school math seriously nor was I any good at that level. I think it'll serve you well considering the level at which you're at, as it did me. I've reread that book four times to date and still enjoy it.
One book I'd recommend is "1089 and all that" (http://www.amazon.com/1089-All-That-Journey-Mathematics/dp/0...).
I've bought copies for friends and family whose maths backgrounds range from school to degree level and they've all enjoyed it.
It's not a textbook, more a conversational account of a few topics the author has found interesting through his life from a young boy to an Oxford don. It's very readable and the author's delight in maths and problem solving shine through
I've bought copies for friends and family whose maths backgrounds range from school to degree level and they've all enjoyed it.
It's not a textbook, more a conversational account of a few topics the author has found interesting through his life from a young boy to an Oxford don. It's very readable and the author's delight in maths and problem solving shine through
Get "The Princeton Companion to Mathematics" - http://press.princeton.edu/titles/8350.html
Seriously? Your reply to "I'm trying to teach myself K-12 math" is "find out what mathematicians think about, instead"? My guess: you are trolling or have never read the Princeton Companion.
I'm suggesting it because, whatever practical tutorials the poster is looking for, the Companion will make for fascinating reading.
When I was in school and plodding through physics and chemistry class, I was wildly excited by other, more advanced books, which I devoured. Ditto for biology.
I wish the Companion was out then, for I would have loved it (although, my love for mathematics wasn't in any way diminished).
Sure, the book is not going to teach you K-12 stuff, but it is almost like standing on the edge of mountain, giving you a vista of the wonderful world of math.
When I was in school and plodding through physics and chemistry class, I was wildly excited by other, more advanced books, which I devoured. Ditto for biology.
I wish the Companion was out then, for I would have loved it (although, my love for mathematics wasn't in any way diminished).
Sure, the book is not going to teach you K-12 stuff, but it is almost like standing on the edge of mountain, giving you a vista of the wonderful world of math.
Thanks for the link to this. Not that I think this book is the best thing for the OP (although he/she may enjoy it), but this is just the kind of book that I would like to read.
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It will only teach you a small bit of math (i.e. Trigonometry) but try programming a little 2D rocket ship game with some physics and movement etc. It can be fun and you will get a better feel for the maths involved. don't use libraries. Use javascript with canvas - easy as pie.
Find a problem that piques your interest, then learn the math that's relevant to the problem. You can choose a problem from most any subject, like math, physics, or economics.
Having a problem that piques your interest--something to work toward--is a good motivator.
Having a problem that piques your interest--something to work toward--is a good motivator.
How would I go about finding said problem? Sorry if this seems stupid. There is no real problem that piques my interest right now perhaps because I don't really know the field all too well. I just want to enjoy learning math and get better at solving problems.
Actually, this is what the book "Who Is Fourier?: A Mathematical Adventure" is about. It's the adventure of a group of families who wanted to learn more about languages, which led them to sounds which led them to learn about Fourier transforms. It's a lot of fun and it actually has real math that you often get to learn in a signal processing class in college. While it is very focused, this book and the approach will pique your interest in other topics.
It depends on where your interests lie. Math is all around us and is involved in most everything we do, so any subject will work.
When was the last time something made you stop and think "that's cool--how does it work?" What was it and what about it caught your attention?
Just live your life and something will eventually catch your eye. Every "huh, that's interesting" moment is an opportunity to dissect it and learn the math that's involved.
When was the last time something made you stop and think "that's cool--how does it work?" What was it and what about it caught your attention?
Just live your life and something will eventually catch your eye. Every "huh, that's interesting" moment is an opportunity to dissect it and learn the math that's involved.
Might be a nice time for GEB. There's even a K-12-ish course for it at MIT OCW:
http://ocw.mit.edu/high-school/courses/godel-escher-bach/ind...
http://ocw.mit.edu/high-school/courses/godel-escher-bach/ind...
The one that made me truely see the beauty of maths:
Hallucinogenic drugs - due to the hallucinations basis in geometric shapes and patterns.
Not really useful for teaching yourself K-12 math however for the beauty of it nothing better
Not really useful for teaching yourself K-12 math however for the beauty of it nothing better
I recommend seeking out translated Soviet math textbooks. They're dense and unforgiving, but are conceptually correct. You could do far worse.
Do you have any specific textbooks I could take a look at?
Hijacking the thread here, but could someone quickly recommend a college-level calculus book?
Spivak's Calculus, http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098... is a good bet. But it's very theory-involved.
let's start with "how to solve it" (g. polya), if you love to learn the thinking of math.