If you're "good" at math (have a working knowledge equivalent or borne of an advanced degree) I've got a story for you, then a question.
Story When I was little my grandma would give me math problems to do for fun. I'd spend hours adding, multiplying, dividing, memorizing the tables grades before I needed. I loved math. Worksheets would be done in a flash and I'd sit around waiting for my classmates to get done. The pace was gruelingly "lowest denominator" (hahaha forgive me). Then high school hit. I placed in the advanced math program. I'd get 0's for arriving at the right answer the wrong way. At the time I felt i needed to stay in the advanced track to keep my parents happy, and also because being in advanced track = smart, and I wanted to be seen as smart. So I gave up concept-based learning and became good at regurgitating, which slowly eroded all the joy I had for it. I'd learn enough to do ok on the tests and graduated having taken calculus I (with a TON of holes in my overall math knowledge).
Since then I've avoided math in college course-wise (except the time i took calc I again in college and got a C), but always been oddly attracted to it. I just realized all of this. I think underneath I love math, so I've decided to pursue it again -- but this time on my own and only as far as my love of it will take me (or not).
Question How can I learn and love math again?
i lied, i have more questions
what's the best way to learn (on my own)?
what are some resources i can use to do this?
how can I know its comprehensive?
and because my nature is results-oriented, are there tests I can take to prove I know it (without having to take the class and stuff?)
go to khan academy on youtube, tons of great math resources there. Get books from your local/uni library and work through those. (lots have problem sets etc, and im sure you can find tests online for certain subjects). I know a lot of schools have open courseware and stuff too, like mit has it for discrete math etc.
You can never know it's comprehensive because there's always more to learn!
Where did you stop your math education, topics wise? I can give you some good places to look if I know specifically what you want to learn. Is calc 1 the only university math course you completed? Usually they force you through some kind of algebra/precalc first.
Also what kind of math do you want to do in your free time? Math puzzles/thinking games or learning new topics? Real world or theoretical? Statistical or emperical? There is so much math out there it really makes finding a starting place hard unless you have a specific idea what you're trying to learn and what you plan on applying it to.
EDIT I was a math major in college and I think the most fun math topics are
1st place for fun is discrete math. A dictionary will tell you discrete math is math that is discrete or "smooth" and not continuous... but that's very vague. It's pretty much a mix and match of different topics... all intuitive enough not to require strenuous memorization yet very fun to think about. The popular topics you learn are theoretical computing (state machines), logic, graph theory, set theory, combinatorics. topology, and some light number theory.
2nd place for fun is cryptography. Learning the standard for data encryption is very cool/awe inspiring. In this sort of class you will learn about many different cryptographic algorithms, specifically RSA which is used to encrypt credit card numbers amongst other things. You will also learn about zero knowledge proof protocols which I thought were so awesome I did my thesis on them... zero knowledge proofs are essentially what internet arguments are. Someone tries to convince someone they know something without being able to offer any proof that they do know what they are talking about, so you must prove your integrity to someone who assumes zero knowledge. It's an enlightening class.
3rd most fun because I have time are algorithms in computing and complexity theory. Complexity theory is pretty much classifying how complicated an algorithm is or how difficult it is to solve... problems such as the Towers of Hanoi and The Traveling Salesman for example are considered "difficult" because as these problems grow more complex, computing them becomes exponentially more difficult as opposed to many problems which can be solved in polynomial time.
I'm going to do maths to avoid now because they are boring and/or pointless and/or extremely frustating (all personal opinions mind you)
1st most painful is Linear Algebra. Everything about this topic is difficult and strenuous. It is memorizing rules, regurgitating, equation grinding, monotony. Do not learn about linear algebra.
2nd most painful is differential equations. It's everything I said about the last one. Imagine a ball falling to the ground. That ball is falling at a velocity, the velocity is acceleration of gravity minus air resistance. Finding the velocity of the ball at any given instant as it is falling is a differential equation, because the ball's velocity is derivative of the acceleration of gravity. If any of that is interesting in the slightest to you, then you'll love differential equations.
3rd most painful is calculus, specifically calculus 2 and integration by parts. Knowing a formula will not help you here. Knowing theory will not help you. If you want to be able to use integration by parts all you can do is practice it, over, and over, and over, because it has very vague rules and there is literally no way of learning other than going through every motion manually. I tend not to like math where a concept is not enough to get a good understanding of a topic.
Check out [AoPS Book #1], any of the AMC12s questions, these are nice problems ranging from anything traditionally below calculus. You can move to the AIME, then the USAMO, if you want dem proofs.
The Putnam exams are an extremely challenging exam involving calculus and higher-level questions.
These are extremely hard, but they're what all the bestestest math people are doing. Worth checking out.
Wow I feel much the same way. When I was little I'd always do math with zeal, but around middle school became completely repulsed by it. I wonder if it was because it got challenging and that ruined my image of being "smart", or if it was just the way it was taught, like you describe. I've been reading Euclid's Elements and some biographies of mathematicians in the library recently to try and rediscover that love of math. It's sad to be closed off from such a deep sea of intellectual enjoyment.
I think you're simply tired of regurgitating information and doing arithmetics, which is what most people think math is. It's a waste to not take college-level mathematics courses since that's where math gets really interesting and bizzarre. Going into college I had no love for math, even though I picked a major in computer science, but after learning a bunch of weird calculus stuff and how math really came to be, I have to say that I can't live without it.
I also highly recommend looking into cryptography like Sinensis said (don't shotgun this though, it requires a fair bit of background knowledge beforehand), as it is one of the more interesting and practical sides to math. Rigorous mathematical proofs also require a high level of thinking and logic, which might be what you're looking for in terms of a result-oriented learning process.
Drop down to lower math courses! You liked math because you were good at it. Now your standards are too high and you are a small fish in a large pond.
Math team.. so much fun in HS.
You probably won't like college math. Try it out on some online courses like MIT opencourseware. At the least you'll know if you'll want to major in math.
I think a lot of entry-level undergraduate math (real analysis, abstract algebra) is relatively boring. Later on, it becomes absolutely fascinating IMO, but the early stuff isn't that interesting.
It's completely possible to just buy textbooks and learn, but this requires a lot of effort. It has the advantage that after the really difficult initial stage of trying to learn this way (which may take a long, long time) you can learn just by picking up a random math book and reading it. I found learning this way extremely rewarding, however, judging from your description I'm assuming you don't have the time to learn this way.
What do you like to learn in math? Do you like hard problems or do you like the theory behind the problems?
If you enjoy solving hard problems with very clever solutions, combinatorics is a field with a lot of those. It has the advantage that you don't need to learn much before you can start doing things with it. It's part of what's sometimes called discrete math, as is noted by the above posts. If you're more interested in theory than hard problems, then you might want to look at other fields (though some subfields of combinatorics do have a lot of theory.)
If you're more interested in theory, I can't think of anything you can really do that is immediately applicable. Abstract algebra felt like needless generalization until I saw its uses later on.
EDIT: @Above: As I said, a lot of abstract undergraduate math is boring. But the later stuff, where you can actually do things, is awesome.
On March 06 2012 15:14 KurtistheTurtle wrote: So I gave up concept-based learning and became good at regurgitating, which slowly eroded all the joy I had for it. I'd learn enough to do ok on the tests and graduated having taken calculus I (with a TON of holes in my overall math knowledge).
I have no idea why you didn't realise that you only hate regurgitation earlier than you did xD.
Here are some that are formatted by year level, you can look up the pre reqs for a specific subject if you cbf learning every branch of mathematics:
Hrm.. have you read the book "Fermats last theorem" by Simon Singh ? Its not so much about learning math, but about the beauty of math (and a lot about history, too). Its a pretty good read imo, and reading it reminded me of a lot of the things I used to love about math. It has a few little problems to solve for yourself too, but that isn´t the major point of the book. Got me pretty motivated in my studies (again).
For the practical side, I´d recommend (like Divinek already did) the Khan Academy Website.
Something I forgot: I've heard very many good things about the book "Princeton Companion to Mathematics." I don't know if it would be applicable to your case, but some people have told me that it's applicable for all levels. Would anyone who has the book here be able to comment?
Wikipedia says "Much of the rest of the book, such as its collection of biographies, would be accessible to a mathematically inclined high school student."
As a math and CS major going through university, there is a lot of pain you have to suck up in the terminology world. Calculus? That's EASY if you understand the rules that you are taught. Maybe it is a multidisciplinary way of looking at a problem, but you have to be able to dissect when X equation isn't valid, why, and how to "proxy" it successfully. (from CS/Physics background).
One of the most fun fields to get into is simple game theory and later math modeling. It is a blend of statistics, psychology (I know right?), theory--yet-applicable experimentation, and simple math. The rigor is in setting up the math that you do actually do. This is the rigor in, well, most any applied mathematics. Setting up the damn problem to use our tools given by calculus, linear algebra, diff eq, etc. (and you can "dumb down" the math if you aren't ready for high level math). This is a good intro to high level math (and if you truly love it). When you truly understand what you're calculating the problems becoming infinitely easier.
On the topic of application--start to program (via Java, easiest to pick-up-and-go precedurally) and try and program equations and tell it how exactly to do some special things. Then try to generalize it (IE derivative rules of a function) so that it covers all exceptions and you know the damn rules inside out between practicing them and then programming each individual part and see each and every action done to get from data to answer. Plus then you know how to program a bit.
On March 06 2012 15:44 Sinensis wrote: 1st place for fun is discrete math. 2nd place for fun is cryptography. 3rd most fun because I have time are algorithms in computing and complexity theory.
1st most painful is Linear Algebra. 2nd most painful is differential equations. 3rd most painful is calculus, specifically calculus 2 and integration by parts.
Hmm, this is an interesting claim. I won't disagree that the former three areas are more "fun" than the latter, but I don't agree that someone interested in math should "stay away" from linear algebra and calculus (which sort of encompasses differential equations).
Linear algebra and calculus aren't really meant to be fun, necessarily; they're important as tools. To an engineer or physicist (or other scientist, sometimes), and of course to a mathematician, they are merely tools, sort of like arithmetic or basic algebra. Both subjects allow us to connect geometric intuition to algebraic statements: in their most basic incarnations, linear algebra relates vectors to solving systems of equations, and calculus gives rules for computing rate of change and areas related to well-behaved functions.
Both are used in optimization problems (though different kinds: linear algebra for discrete optimization, such as "How does an airline minimize the cost of flights between the cities it serves," whereas calculus is for continuous optimization, such as "Why are soap bubbles spheres?"). In general, linear algebra sees more discrete applications than calculus does; algorithms and discrete math (two of the "fun" things) often use powerful ideas from linear algebra. But you could just as easily maximize some discrete-valued function by pretending it's continuous and applying calculus. A classic example is the following question:
There are n people in a room; any pair of people either know each other or don't. Say there are P pairs of people who know each other initially; every 5 minutes, every pair of people with a mutual friend are introduced to each other (and only such pairs are introduced). Eventually, everyone in the room knows each other; what's the minimum possible P such that this happens?
(This can be reformulated as n cities with roads between some of them; P is the fewest number of roads required so that one can reach any city from any other city by driving along roads.)
So basically, don't discount linear algebra and calculus immediately, annoying/boring/repetitive as they might be. That's like saying, "Multiplying numbers is boring! I don't want to learn how to multiply!" ... it's true, but it's your loss.
Also, one of the more interesting areas of math for me is number theory, the study of integers (which, I guess, is a branch of discrete math---the other mainly being lumped into "combinatorics"---and rather related to cryptography and algorithms;RSA encryption is based on a fairly basic [url=http://en.wikipedia.org/wiki/Euler's_theorem]theorem of Euler[/url.) Most "number tricks" also result from some basic number theory and algebra. A lot of number theory focuses on studying primes, because you can build the rest of the integers greater than 1 out of them.
And though we know a lot about primes (such as Euclid's brilliant proof of the infinitude of primes ***), there's still a ton we don't know. For example, a recent theorem, the Green-Tao theorem, states that there are arbitrarily long arithmetic sequences (in which any two consecutive terms have the same difference) of primes. This is incredibly non-trivial; people haven't been able to compute such arithmetic sequences of primes longer than 26 terms (all over 40 quadrillion), but somehow, some high-powered math machinery magically showed that such sequences of any finite length exist. An interesting "dual" theorem is Dirichlet's theorem, which says that any arithmetic sequence containing more than 1 prime contains infinitely many.
Some famous unsolved problems which have proved intractable over many decades, despite being simple enough for a high schooler to understand, include Goldbach's conjecture (any even number > 4 is the sum of two odd primes) and the Collatz conjecture (start with a positive integer n, and consider the following rule: replace n by n/2 if n is even, and replace n by 3*n+1 if n is odd. Show that no matter which n you start with, you eventually get to 1.)
Maybe I'm getting too technical here, and maybe you're not interested in proof-based math (though that's all there is starting at the undergraduate level), but there's a lot of fun stuff in math; you can explore some of the topics people have mentioned, or check out some middle/high-school math contest problems (in the US, these include AMC, MathCounts, ARML, Mandelbrot, various Math League type things, etc.), which are pretty approachable but require serious thought.
*** Proof: Say there are only finitely many primes; call these primes p_1, p_2, ..., p_n. Consider the number p_1*p_2*...*p_n + 1; this number is not divisible by any of p_1, p_2, ..., p_n (because it leaves a remainder of 1 when you divide by any of the p's.) But since it's an integer, it must have a prime factor q which is not any of p_1, p_2, ..., p_n; this contradicts the assumption that p_1, p_2, ..., p_n were the only primes that exist, so there must be infinitely many primes.
Story When I was little my grandma would give me math problems to do for fun. I'd spend hours adding, multiplying, dividing, memorizing the tables grades before I needed. I loved math. Worksheets would be done in a flash and I'd sit around waiting for my classmates to get done. The pace was gruelingly "lowest denominator" (hahaha forgive me). Then high school hit. I placed in the advanced math program. I'd get 0's for arriving at the right answer the wrong way. At the time I felt i needed to stay in the advanced track to keep my parents happy, and also because being in advanced track = smart, and I wanted to be seen as smart. So I gave up concept-based learning and became good at regurgitating, which slowly eroded all the joy I had for it. I'd learn enough to do ok on the tests and graduated having taken calculus I (with a TON of holes in my overall math knowledge).
This has to be the most retarded thing I have ever read in my entire life. Yet, this is how almost all schools work nowadays.
The Khan Academy is actually pretty bad for a this. You can find some stuff there (haven't watched a lot, but Vi Hart's videos are listed, and there are some math contest problems that might be good), but most of it is get-the-answer high school math done the same boring way it's always been done in high school. I'm a big fan of Khan Academy - they're solving a very important problem - but it's a different problem.
You should really, really be doing proof-based math. Exactly what proof-based math it is isn't that important. I agree with the discrete math recommendation, especially combinatorics. Theoretical computer science is a great area to try, as is number theory. As far as I know, nothing on the internet is all that amazing at teaching this stuff. The best thing by far would be a college class. Failing that, get a good book and work through it. The Art of Problem Solving books look pretty good to me, but maybe a little slow. If you're just getting back into it, though, that might be good. College textbooks would usually be best, but they're expensive. If you want to see some cool stuff, Michael Sipser's Theory of Computation book might be a good one to try.
I just don't understand why people call cryptyo "funny". Past the fun with RSA and zero-knowledge protocols, it's boring as hell, you have to do dirty things to be secure and it become pretty hard (arithmetics with a lot of tricks and so). Complexity theory is cool, but when you go past the bascics (turing machines, P,NP, RP) it become really boring too (with really wierd complexity classes that have no algorithm running into :D)
Linear algebra is far easier, and quite interesting (you will be seeing groups everywhere :D). I have been surprised by complex analysis, that is quite interseting (considering a name that could be boring), with powerfull theorem (residue theorem).
On March 06 2012 15:14 KurtistheTurtle wrote: I'd get 0's for arriving at the right answer the wrong way.
Math is not only about beeing able to find the right answer, but be able to back up your claims. There is no "wrong way", just flase proofs :D. For me, pleasure of math comes from finding the answer, and then try to prove it. An unproved answer is generally false, because you always forget silly cases and so. If you want to learn some math, I advise you to take lecture notes, read the definition, read the theorem, and try to prove it by yourself (using your intuition, but with rigourous argumentation). If your proof don't look like the lecture proof and it's shorter, you might have took a wrong shortcut. If it's the other way, you probably missed one. You can have as an goal to have the same proof that in the lecture notes, to train you to have a sound reasoning. Making the proof is essential, because just read some theorem and try to remeber it don't cut it if you ant to know what you are doing, and cut any intuition you could have (by using theorem as blacckbox)
On March 06 2012 20:00 ]343[ wrote: 1st place for fun is discrete math. 2nd place for fun is cryptography. 3rd most fun because I have time are algorithms in computing and complexity theory.
1st most painful is Linear Algebra. 2nd most painful is differential equations. 3rd most painful is calculus, specifically calculus 2 and integration by parts.
I like the part of math where I can take a process, quantify all the interrelated variables, then manipulate numbers to reveal insights or realities which aren't immediately obvious. Find the pockets of value where intuition fails. I hated statistics when I took it (because of the teacher...and the fact it was "math at school") but recently I've been helping my gf & another friend with their homework. When I can manipulate numbers to arrive at an answer, I feel like a boss. If I could do this to massive data sets and provide insightful, relevant and useful insight & information...that would get me up in the morning. In the coming years, many amazing things will come out of the proper analysis & communication of the MASSIVE amount of data now available.
I'm the guy who's naturaly predisposed to do just that. I like internalizing large amounts of information, figuring it all out, finding the valuable advancement or achievement of a particular goal, then moving on to the next thing. I also like spending lots of time alone and not being disturbed. I also love focusing on one particular, giant, intricate problem. This is the skill I want to cultivate.
I don't know what its called though. Applied statistics, "big data" analysis, etc.
What are the classes I should take centered in this area (undergrad or grad)?
I'm thinking: All the statistics courses available and matrix algebra. Maybe it's linear algebra?
Also, I was looking into HADOOP. What other related stuff should I learn? (learning HTML5 already, so what else?) Basically marketable applications of the theoretical stuff I learn so I can apply it right away and stay excited.
EPT
