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I came to a career in software development with a degree in English, rather than Computer Science or another science/engineering background. I have gone a long way on my self-taught basis, but after 10+ years of doing this, I want to go back and fill in the gaps, particularly with the math.
The obvious place to give myself a Comp-Sci education is to go through The Art of Computer Programming. However, as I didn't take all that much math and my last math class in college was in 1995, I need some brushing up and augmenting to even be able to read the math notation in TAOCP.
My thought was to go to Khan Academy and work through the necessary topics as a remedial prereq to reading TAOCP. However, in a Catch 22, I'm trying to figure out which topics do I actually need to go through as prep.
So, what I'm wondering is, if someone basically only had high school math (I've got a bit more than that, but I think it's a valid question for someone to approach this with just high school as a background), what math "classes" does one need from somewhere like Khan Academy in order to start TAOCP prepared to read and understand the included math?
Knuth is not the place to start. It's the place to strive for.
So, remedial math is good. But don't beat yourself up if it takes years to master the math required to read (and understand Knuth).
Old, but still excellent: http://www.amazon.com/Fundamental-Structures-Computer-Science-William/dp/0201087251
Look for titles like this:
http://www.amazon.com/Discrete-Mathematics-Computer-Science-Curriculum/dp/1930190867
Or like this
http://books.google.com/books?id=b9nHPJvP7xgC&printsec=frontcover&dq=computer+science+mathematics&source=gbs_similarbooks_s&cad=1#v=onepage&q&f=false
You want "discrete mathematics" to start with.
Also, you'll eventually need
http://www.amazon.com/Computability-Computable-Functions-Foundations-Mathematics/dp/0534103561
Or something similar.
A very easy to understand book is Discrete Math with Applications by Susanna Epp. Excellent book, great application and interesting. Buy it used. It should provide a good foundation.
Echoing the others, a discrete mathematics class is what to aim for. One of the strengths of Knuth's books is the extensive algorithm analysis in the text and in the exercises. A undergraduate sequence in calculus will be needed to understand some of the analysis. And "Seminumerical Algorithms" would be best appreciated I think with a undergraduate number theory course. Plus number theory is fun in its own right!
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I haven't taken any math classes above basic college calculus. However, in the course of my programming work, I've picked up a lot of math and comp sci from blogs and reading, and I genuinely believe I have a decent mathematical mind. I enjoy and have success doing Project Euler, for example.
I want to dive in and really start learning some cool math, particularly discrete mathematics, set theory, graph theory, number theory, combinatorics, category theory, lambda calculus, etc.
My impression so far is that I'm well equipped to take these on at a conceptual level, but I'm having a really hard time with the mathematical language and symbols. I just don't "speak the language" and though I'm trying to learn it, I'm the going is extremely slow. It can take me hours to work through even one formula or terminology heavy paragraph. And yeah, I can look up terms and definitions, but it's a terribly onerous process that very much obscures the theoretical simplicity of what I'm trying to learn.
I'm really afraid I'm going to have to back up to where I left off, get a mid-level math textbook, and invest some serious time in exercises to train myself in that way of thought. This sounds amazingly boring, though, so I wondered if anyone else has any ideas or experience with this.
If you don't want to attend a class, you still need to get what the class would have given you: time in the material and lots of practice.
So, grab that text book and start doing the practice problems. There really isn't any other way (unless you've figured out how osmosis can actually happen...).
There is no knowledge that can only be gained in a classroom.
Check out the MIT Courseware for Mathematics
Also their YouTube site
Project Euler is also a great way to think about math as it relates to programming
Take a class at your local community college. If you're like me you'd need the structure. There's something to be said for the pressure of being graded. I mean there's so much to learn that going solo is really impractical if you want to have more than just a passing nod-your-head-mm-hmm sort of understanding.
Sounds like you're in the same position I am. What I'm finding out about math education is that most of it is taught incorrectly. Whether a cause or result of this, I also find most math texts are written incorrectly. Exceptions are rare, but notable. For instance, anything written by Donald Knuth is a step in the right direction.
Here are a couple of articles that state the problem quite clearly:
A Gentle Introduction To Learning
Calculus
Developing Your Intuition For
Math
And here's an article on a simple study technique that aims at retaining knowledge:
Teaching linear algebra
Consider auditing classes in discrete mathematics and proofs at a local university. The discrete math class will teach you some really useful stuff (graph theory, combinatorics, etc.), and the proofs class will teach you more about the mathematical style of thinking and writing.
I'd agree with #John Kugelman, classes are the way to go to get it done properly but I'd add that if you don't want to take classes, the internet has many resources to help you, including recorded lectures which I find can be more approachable than books and papers.
I'd recommend checking out MIT Open Courseware. There's a Maths for Computer Science module there, and I'm enjoying working through Gilbert Strang's Linear Algebra course of video lectures.
Youtube and videolectures.com are also good resources for video lectures.
Finally, there's a free Maths for CS book at bookboon.
To this list I would now add The Haskel Road to Logic, Maths, and Programming, and Conceptual Mathematics: A First Introduction to Categories.
--- Nov 16 '09 answer for posterity--
Two books. Diestel's Graph Theory, and Knuth's Concrete Mathematics. Once you get the hang of those try CAGES.
Find a good mentor who is an expert in the field who is willing to spend time with you on a regular basis.
There is a sort of trick to learning dense material, like math and mathematical CS. Learning unfamiliar abstract stuff is hard, and the most effective way to do it is to familiarize yourself with it in stages. First, you need to skim it: don't worry if you don't understand everything in the first pass. Then take a break; after you have rested, go through it again in more depth. Lather, rinse, repeat; meditate, and eventually you may become enlightened.
I'm not sure exactly where I'd start, to become familiar with the language of mathematics; I just ended up reading through lots of papers until I got better at it. You might look for introductory textbooks on formal mathematical logic, since a lot of math (especially in language theory) is based off of that; if you learn to hack the formal stuff a bit, the everyday notation might look a bit easier.
You should probably look through books on topics you're personally interested in; the inherent interest should help get you over the hump. Also, make sure you find texts that are actually introductory; I have become wary of slim, undecorated hardbacks labeled Elementary Foobar Theory, which tend to be elementary only to postdocs with a PhD in Foobar.
A word of warning: do not start out with category theory -- it is the most boring math I have ever encountered! Due to its relevance to language design and type theory, I would like to know more about it, but so far I have not been able to deal...
For a nice, scattershot intro to bits of many kinds of CS-ish math, I recommend Godel, Escher, Bach by Hofstadter (if you haven't read it already, of course). It's not a formal math book, though, so it won't help you with the familiarity problem, but it is quite inspirational.
Mathematical notation is is akin to several computer languages:
concise
exacting
based on many idioms
a fair amount of local variations and conventions
As with a computer language, you don't need to "wash the whole elephant at once": take it one part a at time.
A tentative plan for you could be
identify areas of mathematics that are interesting or important to you. (seems you already have a bit of a sense for that, CS has helped you develop quite a culture for it.)
take (or merely audit) a few formal classes in this area. I agree with several answers in this post, an in-person course, at local college is preferable, but, maybe at first, or to be sure to get the most of a particular class, first self-teaching yourself in this area with MIT OCW, similar online resources and associated books is ok/fine.
if an area of math introduces too high of a pre-requisite in terms of fluency with notation or with some underlying concept or (most often mechanical computation and transformation techniques). No problem! Just backtrack a bit, learn these foundations (and just these foundations!) and move forward again.
Find a "guru", someone that has a broad mathematical culture and exposure, not necessarily a mathematician, physics folks are good too, indeed they can often articulate math in a more practical fashion. Use this guru to guide you, as he/she can show you how the big pieces fit together.
Note: There is little gain to be had of learning mathematical notation for its own sake. Rather it should be learned in context, just like say a C# idiom is better memorized when used and when associated with a specific task, rather than learned in vacuo. A related SO posting however provides several resources to decipher and learn mathematical notation
Project Euler takes problems out of context and drops them in for people to solve them. Project Euler cannot teach you anything effectively. I think you should forget about it, if it is popular it does not mean anything. You cannot study Mathematics through Project Euler as it contains only bits and pieces(and some pretty high level pieces) that you're supposed to know in order to solve the problems. Learning mathematics means to consider a subject and a read a book about it and solving exercices or reading solutions, that's how you learn math. If it so happens that through your reading you find something that is close to some project euler thing, your luck , but otherwise Project euler is a complete waste of time. I think the time is much better invested choosing a particular branch of mathematics and studying that. Let me explain why: I solved 3 pretty advanced Projec Euler problems and they were all making appeal to knowledge from Number theory which I happened to have because i studies some part of it. I do not think Iearned anything from Project Euler, it just happened that I already knew some number theory and solved the problems.
For example, if you find out you like number theory, take H. Davenport -> Hardy & Wright -> Kenneth & Rosen's , study those.
If you like Graph Theory take Reinhard Diestel's book which is freely available and study that(or check books.google.com and find whichever is more appropriate to your taste) but don't spread your attention in 999999 directions just because Project Euler has problems ranging from dynamic programming to advanced geometry or to advanced number theory, that is clearly the wrong way to go and it will not bring you closer to your goal.
This sounds amazingly boring
Well ... Mathematics is not boring when you find some problem that you are attached to, which you like and you'd like to find the solution to, and when you have the sufficient time to reflect on it while not behind a computer screen. Mathematics is done with pen and paper mostly(yes you can use computers .. but that's not really the point).
So, if you find a real-world problem, or some programming problem that would benefit from
you knowing some advanced maths, and you know what maths you have to study , it can be motivating to learn in that way.
If you feel you are not motivated it is hard to study properly.
There is also the question of what you actually mean when you say learn. Does the learning process stop after you solved the problems at the end of the chapter of a book ? Well you decide. You can consider you have finished learning that subject, or you can consider you have not finished and read more about it. There are entire books on just one equation and variations of it.
The amount of programming-related math that you can learn without formal training is limited, but it's more than enough. But maybe you can self-teach yourself.
It all boils down to your resources and motivation.
To know mathematics you have to do mathematics not programming(project euler).
For beginning to learn category theory I recommend David Spivak's Category Theory for the Sciences (AKA Category Theory for Scientists) because its relatively comprehensible due to many examples that enable understanding by analogy and which quickly builds a foundation for understanding more abstract concepts.
It requires the ability to reason logically and an intuitive notion of what is a set. It proceeds from sets and functions through basic category theory to adjoint functors, categories of functors, sheaves, monads and an introduction to operads. Two main threads throughout are modeling databases in terms of categories and describing categories with annotated diagrams called ologs. The bibliography provides references to more advanced and specialized topics including recent papers by Dr. Spivak.
An expected outcome from reading this book is the capability of understanding category theory texts and papers written for mathematicians such as Mac Lane's Category Theory for the Working Mathematician.
In PDF format it is available from http://math.mit.edu/~dspivak/teaching/sp13/ (the dynamic version is recommended since its the most recent). The open access HTML version is available from https://mitpress.mit.edu/books/category-theory-sciences (which is recommended since it includes additional content including answers to some exercises).
There are a few questions around that circle around this question but I feel this is different enough.
I've decided I want to improve the breadth and depth of my maths skills specifically in areas that are useful and/or interesting to programmers.
What topics should I study?
What resources do you recommend (blogs/books/online lectures...)?
I'm looking for easy to consume resources because I'll be doing this in my free time, I don't want to spend days struggling through a dense text but I want to get deeper than the surface. I've read the Yegge article on the topic (and most of the comments) which is useful but I think the voting system here will help me focus on the most useful/best resources and topics.
Edit:
I am looking to create myself a study course that I'll follow over the next few years, I'm not looking to solve a particular problem I just want to learn some new skills that will interest me and may be useful in my career in the future.
Concrete Mathematics: A Foundation for Computer Science would be my suggestion for a book that covers some advanced topics.
For an introduction to Discrete Mathematics I strongly suggest this.
I feel very lucky to have been provided this book from University
Any programmer would do well to have a solid understanding on the undergraduate level of these following math courses:
Calculus (at through multivariate calc)
Discrete Mathematics (absolutely essential)
Linear Algebra (necessary for an understanding of matrices)
Combinatorics (further development of Dicrete maths)
Introduction to Abstract Algebra (this will solidify your understanding of modulo number systems, in particular binary, octal, hex etc.). It also gives a deep understanding of set theory which is ubiquitous in practical programming and the comp sci literature.
This is the fundamentals. If your are thinking about graphics or game programming then you have a whole slew of additional courses in physics, graphic arts, and possibly fluid dynamics. Also Differential Geometry is essential for any real world modeling of motion on curved surfaces.
It's a bit off from your question, but let me suggest the Princeton Companion to Mathematics.
It gives an overview of all of mathematics, so it is more than "math useful to programmers", but it's style is as easy to understand as it gets, and the important parts are in there.
Some time ago Steve Yegge wrote a dedicated article about math for programmers. His thesis is: As a programmer you should learn math but you should do so in different way than in shool/university.
His summary is this:
Math is a lot easier to pick up after you know how to program. In fact, if you're a halfway decent programmer, you'll find it's almost a snap.
They teach math all wrong in school. Way, WAY wrong. If you teach yourself math the right way, you'll learn faster, remember it longer, and it'll be much more valuable to you as a programmer.
Knowing even a little of the right kinds of math can enable you do write some pretty interesting programs that would otherwise be too hard. In other words, math is something you can pick up a little at a time, whenever you have free time.
Nobody knows all of math, not even the best mathematicians. The field is constantly expanding, as people invent new formalisms to solve their own problems. And with any given math problem, just like in programming, there's more than one way to do it. You can pick the one you like best.
Math is... ummm, please don't tell anyone I said this; I'll never get invited to another party as long as I live. But math, well... I'd better whisper this, so listen up: (it's actually kinda fun.)
Sad note: Steve abandoned his blog because of too much aggressive feedback.
If you have any interest in game development, 3D graphics, or anything somewhat related to those, then do multivariate calculus and basic physics. This will help you understand the basic concepts much better. Also, linear algebra will help immensely with all of the matrix/vector stuff you will be doing.
If you are NOT interested in these topics, I would still say study calculus and physics. Why? Solving calculus and physics problems gives you good experience in problem solving and exercises the brain. Programmers NEED to be good problem solvers... that is our job. Concepts you pick up from these courses are things you will keep with you the rest of your life.
MIT and Stanford both have really good online courses for topics such as this. Of course you can't just jump into multivariate calculus without some more basic calc, but MIT and Stanford have resources for your basic calculus classes as well. Basic physics will be a little bit easier to pick up. Again, you can check MIT and Stanford for physics.
MIT OpenCourseWare:
Single Variable Calculus
Calculus
Multivariable Calculus
Physics I: Classical Mechanics
Mathematics for Computer Science
Generally speaking, the applications of math to computer programming are pretty domain-specific - that is, you need to know whatever math the specific program you're writing requires. The only mathematical topics I can think of that are generally applicable to all kinds of programming are simple arithmetic and boolean logic, but I think if you didn't already know those you wouldn't be much of a programmer ;-)
Basically, I would just recommend learning the math as needed for whatever project you're working on. If you want to give yourself a good excuse to learn some new math, start a hobby program that does something mathematical.
As for topics, look at some of the answers here. Recommended ressources are difficult for me to give, I'm German speaking. I would recommend starting with linear algebra and geometry, which you will find in computer graphics. Look at the undergraduate math series by Springer for example.
Number theory doesn't have many direct applications to programming (though there are some neat tricks you can use for optimization), but there are several basic concepts that make cryptology much easier to study.
My number theory class used Silverman's Friendly Introduction to Number Theory, which is one of the best math textbooks I've ever seen. It's very easy to read (the title is entirely accurate about its friendliness), but covers a wide range of topics. Silverman is also an author on my cryptography textbook, An Introduction to Mathematical Cryptography. It's more technical, addresses most areas of cryptography, and provides plenty of references for where to find more detail.
Consider Knuth's Art of Computer Programming series. It can get dense, but it will ground you in the math most needed for programming. I'd suggest going for the available fascicles of Volume 4 early on. These books are not for everybody, but if you find them interesting you will learn a whole lot.
They won't teach you calculus or geometry, which are important in many aspects of programming but tend to be more specialized.
I think you should dive into whatever interests you most and in order to find out what that is you should get some books which cover the facts and offer orientation and some books which nurture your motivation and curiosity. You really have to dive into it to find out, it's a pretty individual thing imho.
Facts / Orientation:
Donald Knuth -
Bronstein, Semendjajew
The Science of Programming -
Data Structures and Algorithms
Motivation / Curiosity:
The Road to Reality -
Fermat's Last Theorem -
Godel, Escher, Bach
Also for motivation on the more practical side:
projecteuler.net
What sorts of math problems do you want to solve? 'Math' is a pretty big area!
MIT has some online courses, but that's probably a big time investment.
Wolfram has some tutorials, but again, you need to know what you're looking for.
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It has been discussed on this site before about the relationship between math and programming, and whether one is a subset of the other, etc.
In my recent study of programming, I've found myself more and more wishing I was better at math. You all know the scenario when programming books start to generalize something in a math way ("Therefore, we may say that for all <some single letter>, <lots of letters>"). My eyes glaze over in such situations. I know that that is mostly due to me being stupid, but it seems that if I could just improve my higher math skills, maybe I could get more out of such things.
Major question: Is math indeed something one can "get better at," or is your brain kinda either wired for it or not?
Important follow-up question: If the answer to the above is yes, then what are some ways to go about it?
I think anyone can get better at math. You just have to be determined and practice.
Part of the problem is that math books tend to be written by mathematicians who ceased being math novices decades ago. What you want is books geared to your level and which contain material you can work with.
Some recommendations:
If you can find a copy, get Mathematica and a good book on it (the Schaum's outline is actually pretty good and cheap). I use it all the time to visualize things.
As a programmer, you probably want to aim more for discrete mathematics than calculus.
The Concrete Mathematics book mentioned elsewhere is excellent.
Most introductory discrete math texts have good coverage of the things like logic, sets, combinatorics, probability, graph theory, etc. My school used Rosen's text which I liked.
Linear algebra is useful if you are going to do 3D graphics programming. Most intro texts for engineers will teach you what you need to know. Linear Algebra Done Right is probably the best on "real" linear algebra if you want something more theoretical.
Look for books by Martin Gardner and play with his puzzles. He's an excellent writer and teacher.
Remember that math doesn't change that much. You can get used books for cheap on Amazon and in used bookstores. I always look for the n-1 version when I buy textbooks.
When you first started learning English all those "symbols" (letters) looked like gibberish to you. I'm sure that at some point you were frustrated over your lack of understanding. But slowly, and gradually you began to understand them.
Eventually you were able to construct your own words and sentences using these symbols. After being corrected on it's structure and grammar for years you now have a command of the language.
Math is just like that. Your eyes glaze over because you haven't learned the language. Maybe in school you didn't particularly enjoy math because you didn't see any practical applications for it. Certainly the way we teach math to our students is atrocious, so it is no wonder why many get through school without being well versed in it (for further reading, check out A Mathematician’s Lament which discusses how horrible our current method of teaching math is).
However, it is never too late to get up to a degree of proficiency that will allow you to read many academic Computer Science writings. Start out with Pre-Calculus at your local community college at night (to brush up on everything you have forgotten). Then move on to Calculus and after that take Discrete math. Honestly, this is all the math you will need 99.99% of the time. In less than 2-3 semesters you can be fully caught up and you'll no longer have your eyes glaze over when reading something with some mathematical roots.
I can share my experience...
I have been terrified of math since grade school. Hated it, didn't get the point, didn't pursue it.
By contrast, I have always been fascinated by computers. I study programming from a "need to know"- I can't stand not understanding computers and programming from the lowest to highest levels. I am almost completely self-educated, and have a career as a programmer/architect.
Last year, at my wife's urging, I began to go back to college. I signed up for a remedial Algebra class, knowing it was going to be a pain. It wasn't.
Somehow, through all the years of learning to develop OO software, it seemed that I had tricked myself into learning how to think mathematically. The concepts just weren't that difficult anymore. It may be that I had learned to think in terms of complex systems made up of smaller, less complex ideas.
I am now researching game development, and that is some seriously math-oriented programming. WAY more so than the business development I've been doing to this point. However, I don't find it so daunting because it's applied mathetmatics. Working to solve practical problems seems to make the study less tedious and far more interesting. I have found Wikipedia and Wolfram's Mathworld to be helpful. If you already know how to program, you're ahead of the game learning math.
I'd say it's certainly something anyone can get better at. It takes time and patience and some texts are obscenely dense as far as the notation involved, but if you're willing to put the time in, I think it shouldn't be too horrible.
I'd check out Wikipedia's list of mathematical symbols, and keep if nearby whenever you see a large blob of symbols pop up. Translate them one at a time and put them together in the way that makes the most sense to you (or ask us a few times until you get the hang of it).
It's both. You can get better at math. But you're also indeed limited/endowed by the particular wiring in your brain. What that means is that most likely you can improve your current mathematical skills. However, because of your mental hardware's limits, you may never discover a new theorem.
And when it comes to improving, I think the way as always is to practice. To read mathematical literature, to try to solve mathematical problems and eventually, develop an outlook where you are able to break, as a matter of habit, real-world conundrums you see before you down in mathematical terms.
As for programming's relation with mathematics, I think there's a pretty strong one. In fact, one could argue that a program is nothing but a proof of a theorem, the requirements document being all the inputs to the proof.
Your skills get rusty if not used and knowledge fades with time if not used.
If you don't use your math skills you soon have no math skills. Continuous new learning and practice of the skills you already have will lead to you one day being a math/programming master.
Project Euler has lot of math problems that can only be solved through programming. The problems get more difficult but build on the skills and knowledge acquired in your previous solutions.
I also buy some interesting textbooks at second hand book shops. Their cheap and slowly your skills improve. I use them in conjunction with MIT Open Course ware.
A fun way to practice math is http://projecteuler.net/.
Although it's less systematic/effective than doing a course or reading a textbook.
I know exactly how you feel. I've always wanted to learn more Math, but as I was unable to do it at college after school (not enough space) and not able to take it at university (not able to with a CS degree) I'm still yet to study Math formally since the age of 16.
Math is something that anyone can learn. Some will argue that it gets harder with age but I've met people going on for 60 that are taking Math classes with ease. There's one woman at my university that's going on for 70 and she's a few months off of graduating with a degree in a Mathematics related field. If you want to learn Math then now is the right time, although I'll be the first person to say that it is not easy. Whilst you'll find many of the problems extremely easy with programming experience you'll still find that going through a set of problems takes a lot of time out of your day. I almost finished the MIT OpenCourseWare course on Linear Algebra, then ended up getting a new part-time job, working 10 hours a day, 7 days a week, and forgetting the majority of what I had learnt.
That being said, if you have the time and true dedication I can recommend some links to video lectures that may just help you get on your way.
Pre-Calculus Algebra
Pre-Calculus
Calculus
Applied Probability
Introduction to Statistics
Differential Equations
Linear Algebra
I'm not saying that this is what you need to know. This is what I've set out to learn myself before I graduate from my CS degree, so feel free to pick and choose whatever you feel is best for you.
Studying math is like getting a classical education, one specially suited for programmers and other computer professions. And math's the sort of thing that you can appreciate more as you get older. You realize that it's not about grinding out answers so much as it's about thinking deeply and conceptually. The "answers" you might grind out make a lot more sense that way.
At one time, I would have recommended taking a geometry course, and take some time to learn how to prove theorems, see how the concepts flow together. These days, though, I'd say it may be better to take a course in discrete mathematics. It's much more practical, and there's a lot more variety, but there's still enough theory in there to make it challenging if you want.
Discrete math also provides you with programming challenges you might not have thought of before. Maybe you can hack up a good heuristic to solve an NP-complete problem, like an N-city Traveling Salesman problem. Maybe even come up with a couple solutions, and test which ones work best in which circumstances.
(I never took CompSci classes in college. You can probably tell.)
Go to the local community college and sign up for Calculus 1. This covers functions in the mathematical sense, and has a rigorous refresher course on algebra, and will use just enough of the symbols to get you ahead.
First of all, I would recommend Steve Yegge's Math For Programmers. It pretty much sums up your struggle.
And now I would like to tell a personal story. I was a double major in Math and CS. I learned a lot in the Math classes, but I honestly didn't appreciate it as much as I should have. I will tell you that a lot of things that I did have helped me in my programming career. And it's not about some formula or knowing calculus, or any of that stuff. It's that a solid Math background teaches you how to think in order to solve a problem. To me, that's the Math that you need.
1) Yes.
2) Explore mathematical questions which sound interesting. Buy/read books that give you the information needed. Repeat.
It can absolutely be learned. I personally had the most benefit from the math (especially proof) courses I took in college.
Recommended courses:
Discrete Math
Mathematical Thought
Abstract Algebra
any other proof courses
Recommended book:
The Nuts and Bolts of Proofs, by Antonella Cupillari
I strongly recommend trying to take one or more of these courses at a school of some sort. Find a local college and audit a course.
A 2nd vote for Lockhart's "A Mathematician’s Lament", which recommends that math be taught like painting, poetry, or music -- not for it's practical usefulness but for simple pleasure:
There’s no
ulterior practical purpose here. I’m just playing. That’s what math is— wondering, playing,
amusing yourself with your imagination.
Look at the diagrams in a recent Knuth paper, Dancing Links, and tell me he wasn't having fun making those.
Part of the problem is that a few mathematical symbols convey a heck of a lot of information. If you are reading a normal programming book, it is full of words and code. Neither of these is super verbose (although I often have to slow down much more on code than normal words). However one complicated mathematical equation can easily be a screen full of programming code or words. We have all sorts of simple notations that convey complex processes.
Another issue is that notation is standardized but not exactly. Different books use slightly different notation so it takes a while to get used to it. Also many textbooks leave out key steps in mathematical proofs or even examples. Sometimes even college professors puzzle themselves with the missing steps in a given proof in a textbook and then give their own proof, or give a slightly modified one over the one in the book because they learned it differently or can't recreate the missing step exactly which takes the proof in a different direction.
So anyway just because your eyes glaze over doesn't mean you have to give up. The first time you see the equations you will probably be in read the english text mode and have to pause to consider them. Going over them slowly and paying attention to what all the symbols mean one step at a time may yield the answer for you. If there is some notation you haven't seen before, there is probably an intro chapter or appendix explaining the notation, so check there. Finally, look for other sources. Use google/wikipedia to look up equations for the concept and you may find a derivation and/or proof that you can follow. Additionally the other one may help you to understand the current proofs/derivations better. Even if your understanding of the proof/derivation does not improve, your additional research will probably aid in your understanding of the equation.
I think there are two things to learning math:
1. Learning the general techniques. Ie how to add two fractions, how to differentiate, integrate.
2. Learning to problem solve and apply math to the real world.
I think by picking up math textbooks yo will learn 1. Many math textbooks are organized by section where there will be a few pages showing you a technique and then a bunch of problems. The problems tend to be related to the technique that you just learned and very similar. Ie a section on logarithms will have all problems on logarithms and probably won't include any polynomials. By doing the problems in the section you will learn the techniques. The more problems you do the faster you will get and the more you will understand the concepts. Many times you will find if you work through the problems without explicitly memorizing the formulas, you will find that after you do enough the required formulas will be implicitly memorized. Ultimately if you are having trouble looking at probability formulas you will want to read a probability book. If you are having trouble with sum notation you will want to consult that section of an algebra book, etc...
To learn 2 I think math textbooks don't help as much because each section tends to have problems related to that section. Occasionally there are a few "mixed review" problems or a "chapter review" which mixes problems, but they are typically far in between. Science textbooks like Physics, Biology, Chemistry, etc. tend to be better for this. There you often read the problem, lay it out, and end up using a variety of mathematical tools to to solve it. Sometimes calculus, linear algebra, and geometry all within the same problem. The value here is that it teaches you to problem solve. Generally the SAT/GRE do not test if you know how to do Algebra, they test if you know how to apply it to the real world, and the science problems really help you here. Also programming in general is about problem solving and the better you get at problem solving the better you'll be at programming. Basically in programming you take problems, create a mental model, design a solution, and then model it in your programming language of choice. This is similar to say Physics. You look at the problem, extract a mathematical model, design a solution, right down some equations with the model of the solution, then plug numbers in. I highly recommend physics because after my college physics class word problems became simple for me and they used to be quite difficult (though not impossible).
In day to day programming you probably won't use more than algebra and logic (for if statements and loop conditions). There are some places that use high math like computer games, cryptology, data mining, etc. but for a typical business application you probably won't use more than algebra and logic and maybe a bit of set theory (the stuff so basic you already internalized it). Even in places that use high math (like financial companies) often the business users (or some industry literature) will have done the higher math and you will just need to implement the equations (with some algebra). I only mention this because most programming books don't have more than algebra and logic either, unless you are reading textbooks on Algorithm Analysis (Introduction to Algorithms), Artificial Intelligence, or some other research area. General application books on how to do things are usually short on math.
But depending upon what you are reading math can help. For most computer science algebra + discrete math should be enough. Couple that with some physics practice and you should be good to go. It may still be a slow go but you should have the proper background.
I like combinatorics and algorithms - having fun you learn faster.
study study study!
wikipedia is actually a fairly good math reference. start with something you're interested in learning and follow links until you understand all the building blocks for that initial thing.
practice practice practice!
Schaum's outlines are good for this. If you're interested in probability (which touches on combinatorics), see 50 Challenging Problems in Probability.
Short answer:
There may be people who are to stupid to get good a maths. But those people generally are to stupid to program, too.
So if you have some skills in programming, you might consider yourself smart enough to learn maths as well.
Note: I know there are smart people with a serious math learning disability, but I think thats more like an exception.
I would also recommend project euler While it does not exactly teach math, it gives you problems that you can then look up how to solve. I've always preferred solving actual problems instead of just learning theory.
Math skills are becoming more and more essential, and I wonder where is a good place to brush up on some basics before moving on to some more CompSci specific stuff?
A site with lots of video's as well as practice exercises would be a double win but I can't seem to find one.
It depends on your math level. You should start by revising what you should know till that moment and then go further to algorithm mathmatics, geometry (transforms and etc), statistics and more.
There are tons of places on the internet were you can learn:
http://www.math.cornell.edu/Courses/courses.html
http://ocw.mit.edu/OcwWeb/web/courses/courses/index.htm
http://mathworld.wolfram.com/
and the list is open.
I recommend Project Euler if you want to train number theory and discrete maths. Lots of fun exercises, though you need to know a bit of programming.
Steve Yegge had a good blog post Math for programmers
Quoting some of it:
"But a few things I've learned recently might surprise you:
Math is a lot easier to pick up after you know how to program. In fact, if you're a halfway decent programmer, you'll find it's almost a snap.
They teach math all wrong in school. Way, WAY wrong. If you teach yourself math the right way, you'll learn faster, remember it longer, and it'll be much more valuable to you as a programmer.
Knowing even a little of the right kinds of math can enable you do write some pretty interesting programs that would otherwise be too hard. In other words, math is something you can pick up a little at a time, whenever you have free time.
Nobody knows all of math, not even the best mathematicians. The field is constantly expanding, as people invent new formalisms to solve their own problems. And with any given math problem, just like in programming, there's more than one way to do it. You can pick the one you like best.
Math is... ummm, please don't tell anyone I said this; I'll never get invited to another party as long as I live. But math, well... I'd better whisper this, so listen up: (it's actually kinda fun.)"
I will be boring and recommend actually taking university courses in math.
Without lectures and lessons with an assistant I know I would never be able to learn as much as I have. I just need some kind of motivation, since higher math is really hard.
That is, if you are looking for quite advanced stuff and actually want to get a deep understanding and don't want to crunch numbers. Crunching numbers is why we have MATLAB ;)
It would be good to know what level of math you have, and what you want to do with it. But I guess calculus, linear algebra and discrete math are the most useful courses to take.
I suggest books with good tutorials throughout if you're unable to partake in a maths course. For computer science-related maths Don Knuth's Concrete Mathematics is meant to be very good.
Obviously nothing can replace a good teacher, but good tutorials can come pretty damn close. You really get to learn the subject in the tutorials I think.
Get some videos from www.aduni.org
Math courses
It's a couple of years since this question has been asked, but there are a number of new sites and resources available now:
Khan Academy was originally intended for schoolkids, but it has since expanded to include material that would not be out of place in first-year university courses. It serves as a great way to review and fix fundamentals. It has videos and practice exercises, and keeps track of your progress.
EdX is an evolution of initiatives like MIT Open Courseware. It's now an alliance of universities like MIT, Berkeley and Stanford that offer free online university level courses, with video instruction and learning materials. My only complaint is that some of their courses have prerequisites (like single-variable calculus) that you need to pick up elsewhere, like Coursera, or the original MIT OpenCourseWare site.
Coursera offers more courses than EdX, and many of them are more basic, covering topics like pre-algebra and pre-calculus. The learning interface is not quite as cool as EdX's (which offers a scrollable captioning interface alongside most of it's videos), but the broader range of topics and courses covering fundamentals offers learning you just won't find on EdX.
A lot of the universities will actually publish their lecture materials online. So all you really need to do is find a suitable subject and then read the lecture materials and do the associated work. If you were really sneaky you could probably also go to the tutorials to get help :P
BetterExplained.com has some great math lectures. Its not video lectures but the author gives easy-to-understand explanations on math concepts.
Don't forget that iTunes now has available a load of maths lectures (and other subjects) from various mainstream universities - and all for free.
Since you want to brush up your math
I would suggest you to do a G search on UCCS math online
Or follow this link , and after registering yourself free you can browse the archives
I must say that It's common that you will find people recommending course X .
But rarely will you find people completing their recommended course ..
SO IN the case of number theory you must go for the latest course , the last offering has not high quality video ..
Also for Discrete Math ->There are no lecture notes on this site
So you have to figure out how to establish correspondence two online course (6.042 has good P sets and Notes) And The above Math course for Discrete Math .
I would discourage you to use YouTube (x minutes ) tutorials , Because most of them cover Math like History ..
A good course can be found by G searching Harvard OlI--
It has probability (Non Continuous) - There are P sets without solutions ..