How can I read numbers and mathematical symbols comfortably, at a university level? [closed]

Could it be that you're simply expecting too much? Mathematical formulas are designed to cram a lot of information into little space, but you still need to process all of that information, so you should expect your reading speed to drop dramatically each time you encounter a formula -- at least as measured centimeter by centimeter.

Furthermore (and I can only speak for myself here, but I think it is reasonably universal), there isn't really any spoken form of mathematics. We can read formulas aloud, but then the sounds we make represent symbols on paper -- in contrast to ordinary language, where the symbols on the paper represent sounds. If somebody speaks a formula to me, I have to reconstruct how it looks inside my head before I can begin to understand it. Formulas are a very visual language, with their meanings defined by how symbols appear side by side, or above and below each other, or surrounding each other. You should try to understand them by building some kind of visual model of the computation they describe, not by translating them into words.

I'm stressing this point because it sounds like you're panicking when you come across a formula because the little voice in your head that speaks aloud what you read goes silent. That's normal; it doesn't mean that you're missing some critical ability you're supposed to have. It just means that you need to treat the formula as a picture rather than words, because that is what it is. It's a picture that's sometimes made out of letters, but that's modern art for you -- you may need to approach it visually all the same.

And just as a picture is worth a thousand words, you should expect to spend as much time digesting each formula as it would ordinarily take you to read several paragraphs of ordinary text. It gets a bit quicker than that with practice, for some formulas, some of the time, but you shouldn't feel dissuaded because that doesn't happen instantaneously. Practice takes time.

Also: cheating is allowed. Very often, large parts of a formula are identical to large parts of a previous formula -- and the only thing that really matters is how it differs from the previous formula. In those cases it is expected that you'll just think to yourself, "oh, this thing is the same as that thing over there", without bothering to understand or remember exactly what "that thing" was in detail. You can just compare the relevant parts symbol for symbol without thinking.

In fact, this last point is part of the reason why formulas are designed to be compact and dense with information. It increases the chance that you can keep the entire formula in your visual short-term memory as you move your eyes from one formula to the next, and thereby make it easy to spot that some parts of them look alike, without even being conscious of the individual symbols. (Who knew all those inane "find 5 differences between these two drawings" problems you find in kids' magazines actually train a highly relevant mathematical skill? They do!)

I knew people who had the same problem like you. They went through highschool thinking they don't need mathematics, and then they go to a university where mathematics is a prerequisite.

Mathematics is easy only for those who learned it all the way through middle school and highschool. It is not possible to learn all mathematics in three months. I don't advise you to read any books for introduction to mathematics, because you wouldn't have the time to advance to the level you need to get the exams.

I advise you to do the following:

• first of all, do not get through all highschool math as you say, because you can't do this in 3 months, and you won't have enough time to learn what you need for your present exams
• go to your courses and take notes; do not read books for the exam(unless the teacher says you should, and if he/she does then be sure to ask what parts you need to learn), because books always contain more material than the actual course. Learning the theory by heart can be done easier (I'm surprised to say that, but I've seen it in some of my colleagues) than solving problems. It takes some effort to learn it, and if someone can explain to you what you do not understand, it's even better.
• I guess you will have a problem part in your exam. Take all your seminaries notes, and see what types of problems you need to learn. Before you can proceed to problem solving you might need to learn some trigonometry formulas, differentiation and integration formulas (if you need them), etc (at least make some lists with them; you will learn them as soon as you proceed doing exercises). Solve the seminaries problems by yourself, and only if you get those, search for problem books
• After solving the problems in the seminaries (and homework) you may ask your teacher where you can find additional exercises. Math cannot be learned by reading or memorization. The only thing which can teach you mathematics right is lots and lots of practice: exercises make you understand the theory. More exercises make you feel more free in the field you study. I guess you have other things besides math that you need doing, but reserving a period of time during the day, let's say an hour, for preparing for math can get you on your feet before the exams.
• It is always better if you have someone to explain the theory and problems to you. Maybe you have a friend who knows math and can help you. Ask him to let you do some problems together. An experimented teacher might be better, but maybe you don't want to spend money on private lessons.

Do not despair. 3 months is enough to learn math for your exams. Just be sure you know what you need to learn, so you don't go into unnecessary details, and you are determined enough to succeed.

Another point I want to mention is that if you are unable to understand the concepts you need to learn (for example the basis in linear algebra) it does not mean that you cannot learn to do the problems correctly. I had the same problem in my first year(and so did many of my students I was teaching): I got all my exams with high scores, but didn't understand what I was doing for some time. Now I think I could teach those courses right now without having any notes on me, because by exercising I began to understand more and more what I had learned before. It might be easier for you to learn some kind of algorithms for solving certain types of problems, instead of trying to adapt yourself to every problem you encounter. After you know how to solve algorithmic problems, you will be able to 'improvise' your solutions.

In short: focus on the important stuff, don't get lost in the details and don't let the time go without working because you won't get it back. Hope this helps.

I like this question.

Though Beni briefly touched on it, I think the importance of practice needs to be emphasized and redirected. I, unlike Beni, think reading your math textbooks is a great idea. But as you said, reading math is difficult. Thus it requires lots of practice. Moreover, reading math needs to be active. As Paul Halmos said:

Don't just read it; fight it! Ask your own question, look for your own examples, dicover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?

If you do that, you'll be in there like swimwear.

Don't be afraid to take a lot of time. Taking time allows the concepts to ruminate allows and prevents stress. Nothing inhibits my learning like stress, so preventing (and eliminating, if necessary) it is my #1 priority.

First:

I had quite a poor schooling experience (yes, I am in a third world country)

I'm in the UK and I had a 'poor schooling experience'. It doesn't really matter where you are in the world - if the education system you're in isn't suited to your needs, then it isn't going to serve your needs in the most appropriate way. So don't feel disadvantaged by where you are in the world - people who are bad at maths can be found all over the world!

With regard to your experience with maths, I am in the same position. I am currently studying for a BEng (Hons), but I need to do quite a bit of catch-up work, even covering stuff that was taught at school (albeit, not at all well). I've bought some Open University (http://open.ac.uk) course materials that will help me to do this. In my experience, the Open University has some of the best material available for learning on your own (it has to be!). It's clear, concise and well-organised.

Before I was able to start using that material, though, I was using the courses on the Khan Academy site. Aside from the quality of the short videos, what's particularly good about the Khan Academy site is that it provides a structure for you to learn (a learning path, as I call it) and has practise materials for you to test your understanding.

I've spent a long time looking for good maths learning materials and Khan is by far one of the best I've used as someone who is returning to maths. I'd suggest persevering with it, as you're already using it.

What I would say, as a fellow maths learner, is that you should try to get in to the habit of testing your knowledge and understanding fairly often, to the point of tedium, because only then will you begin to have confidence in using what you know. I have a tendency to learn about something, do a few test questions and then move on. This is no use at all to me, as the very next day I will have forgotten what it was that I learnt! This is my current challenge - finding test sheets of questions/ problems for me to solve on a regular basis.

If all else fails... When it comes to trying to understand maths - don't. I'm of a particular mind that I need to understand the 'why' of something. This can complicate things enormously, when all that's required of you in many circumstances is to just know the 'rules' for a problem. Just accept the method of solving a problem rather than spending your time trying to understand how a problem is solved - it will only frustrate you further (as it does with me!).

I understand the pressure to try to race through what you need to learn, particularly with deadlines looming. Personally, I would want to know what was the minimum required to pass any particular maths course and aim just above that as my threshold. I would also want to know what the syllabus is for the maths course and try to tailor my learning towards that). You might want to look at something like MIT's Open CourseWare programme (http://ocw.mit.edu/) to see how maths fits in to a BSc-level Computer Science programme.

There are many excellent suggestions above, many of which stress the face-to-face element which might be the most help to you at present.

Once you feel you are ready to read textbooks, I recommend the following:

• Maths: A Student’s Survival Guide by Jenny Olive, second edition, Cambridge University Press, 2003.

This is a gentle review of high school mathematics that prepares you for university courses. It has self-test portions that let you know where you stand, highlighted boxes to emphasize major points, clearly identified pitfalls, plenty of examples and exercises, etc.

• How to Think Like a Mathematician: A Companion to Undergraduate Mathematics by Kevin Houston, Cambridge University Press, 2009.

Once you have worked through the previous book, and are ready for university level mathematics, this book should help. There is less "hand-holding" and more of traditional mathematical pedagogy, but all concepts are clearly and concisely defined. Interestingly, Chapter 2 is entitled "Reading Mathematics" which seems rather appropriate to your original question.

• Guide to Essential Math: A Review for Physics, Chemistry and Engineering Students by S M Blinder, Academic Press, 2008.

This book covers material similar to Houston at a brisk pace and with applications in view.

All three books seem to me to be written by excellent teachers who have the ability to communicate mathematics clearly and comfortably, and that might be what you are ultimately seeking.