# How can I learn to “read maths” at a University level?

When I look at math, it's like my mind goes fuzzy.

The only way to describe it is in terms of what I can relate it to. You know how when you read, you see the letters and words, but your brain picks up on the meaning?

When I see numbers, mathematical symbols and equations, I find it incredibly difficult to understand them. Its like trying to read Kanji (without knowing Japanese).

Something simple like $4+24$ is easy, but as soon as I see $\frac{\sqrt{a 2.7 + b 8.6}}{34}$ my brain freaks out. I struggle to read it in my head in a way that makes sense. I literally read it as "a line of numbers".

I am now doing a Bachelor of Computer Science degree part time, and scoring 75% or ABOVE for every subject, except math, which I have failed this semester with a current average of 32%. I had quite a poor schooling experience (yes, I am in a third world country). I don't even know the difference between Calculus and Trigonometry, I completed an access course to get University admission, but the fractions and simple math in the access course were nothing compared to Linear Algebra and Set Theory.

I have about 3-4 months until I retake my uni math courses. My idea is to try and get through the entire math high school syllabus on a site like Khan Academy, and hope that it gives me the grounding and background I need to succeed at a university level.

I'm not sure what to do, I'm desperate for a degree, and more than willing to earn it, it just seems like a huge mountain to climb and I don't know where to start.

The way I see it I have very limited options if I can't "get math". I'm open to suggestions, but Im hoping there is a definitive answer. My question is, in light of the above information, where should I start the learning process if I want to be able to achieve a Bachelor's Degree in Computer Science?

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I think you might want to try a tutor. I mean, it doesn't have to be one that you have to pay. Why can't you ask one of your classmates to help you out? The mountain to climb might turn into a tiny hill. – Rudy the Reindeer Oct 20 '11 at 12:43
How would you try to read Kanji? – Phira Oct 20 '11 at 14:06
or you could look around nearby universities for graduate students who would otherwise be TAs – T. Webster Nov 2 '12 at 4:33

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!)

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I agree with everything except your "spot-the difference puzzles are inane" bit. :) – J. M. Oct 20 '11 at 1:20
(For full disclosure: My personal little internal voice does not actually go silent when I reach a formula -- but it might as well, because it is a complete moron. It just rattles off all symbols it recognizes with no sense of meaning or grammar, such that $x^2$, $x_2$, and $\frac x2$ all sound exactly the same. It also pronounces all of the Greek letters like the Latin ones that look most like them -- $\pi$ sounds like "en", and it only gets worse from there. Useless.) – Henning Makholm Oct 20 '11 at 2:47
HA! My personal little internal voice is classically trained. Also completely useless, but at least it pronounces the Greek letters properly :) – drxzcl Oct 20 '11 at 9:19
You can just stereo-look at the two drawings and the differences will jump out into the third dimension. – Phira Oct 20 '11 at 9:58
Thank you, I have been putting a lot less pressure on myself recently to understand formulas quickly, and it has helped a lot. I think part of the problem was that I expected too much. – anon Sep 11 '12 at 1:05

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.

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I don't want to be understood wrong. Reading math books is great, and necessary from a point on, and I love doing that. But this is not the case for someone who wants a degree in Computer Science, and just wants to be able to pass the math exams with good scores. For someone without experience and without great goals in mathematics reading and learning more than he should can be a burden. – Beni Bogosel Oct 20 '11 at 9:18
@BeniBogosel I think learning to reading math books is useful for anyone because then reading nearly any non-fiction book becomes much easier. Besides, the asker didn't specify what type of CS he's doing. Lots of CS is math. – Quinn Culver Oct 20 '11 at 12:50

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.

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Thank you, it is quite a relief to know there are other people in a similar position. I have had a look at the MIT Open CourseWare, it helped to an extent, but there was a lot I just didn't understand, which is why I was considering fleshing out the background info. – anon Oct 20 '11 at 4:10
Well, I wouldn't expect to understand much of the content, but I'd use it to give yourself an idea of the syllabi and the level of understanding that might be required of you - then you at least have some ideas of what you need to aim towards. – James Oct 20 '11 at 15:57

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.

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I can't disagree with this advice more. Go back and learn what you are missing. So many students can't learn linear algebra because they can't do fractions. Your teacher won't help you with this; you should go back and learn fractions. Also, the book is your friend. The index of the book is also your friend, you should learn how to use it. – Morgan Rodgers Jul 21 '15 at 21:01
I don't get your objection. – Beni Bogosel Jul 21 '15 at 21:41
You say don't go back and learn the fundamentals you didn't learn the first time through. I object. You say don't read the book. I object. You also seem to be saying it is unimportant to worry about the concepts. I kind of shudder to think of someone programming mathematical algorithms without really understanding the concepts. I think my advice would be basically the opposite of yours. – Morgan Rodgers Jul 21 '15 at 21:48
You are free to disagree with me, but you need to be objective. Someone who studies computer science does not have the luxury to spend days/weeks reading math books. You need to use shortcuts, and sometimes those shortcuts work very well, and later on, you even start to understand the concepts. Since you seem to believe that my initial struggle with linear algebra had something to do with fractions, I tell you, I did not have any problem with any aspect of elementary maths back then. It was just something which became clear later. – Beni Bogosel Jul 22 '15 at 10:02
Was referring to the original poster's comments that he feels he is lacking in the fundamentals; maybe this is not fractions, maybe it is elementary algebra. But I think it's smart to go back and learn the things you will be expected to know in a more advanced course. I don't know what your struggles were with linear algebra; I was only speaking about the advice you offered, as it related to someone in the OP's position. – Morgan Rodgers Jul 22 '15 at 14:51

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.

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Thank you so much, I really think this may even be a better approach than what I wanted to do. I have found a good, clearly worded book invaluable to topics in which I do succeed, so I can only believe it would be the same for Mathematics. – anon Dec 9 '11 at 12:39
Checked out the Olive book. Really nicely written. Exercises after every sub-section such that no increment in the reader's knowledge is left untested. Thanks for the rec. – chb Mar 11 '13 at 19:34

This probably isn't the answer that you want to hear, but in my experience, it's the truth: you are probably an entire remedial math sequence (usually 2--3 semesters if not more) away from having a prayer at understanding college-level mathematics courses (ie Calculus I and above).

The good news (if you can call it that) is that the remedial math sequence at most universities covers middle school and high school mathematics (some places even have the equivalent of elementary-school-level remedial math courses).

Now, as to your chances of being able to cram all of that into three months? Well, it might be possible, but you will have to spend nearly every waking moment of every day breathing, sleeping, and eating elementary mathematics. Do every single problem in every (level-appropriate) book you can get your hands on.

You seem to indicate that you're already in a university, so it may very well be too late to retake any kind of "math placement test" that they have (ie an exam to determine whether or not you have to take remedial mathematics). If you succeed in cramming all of this learning into three months (a very difficult feat), you can try to petition the math department to give you an opportunity to test out of any remedial courses.

What you're asking about is possible, but make no mistake, you're in an analogous position to an illiterate person wanting to take a first-semester English Composition course and wondering if he/she can learn how to sufficiently read and write in three months.

Edited to add: My experiences (both as a student and as an assistant professor) were entirely in American schools and universities. Since you may not be attending an American university, they may very well implement remedial math education differently (or possibly not at all). YMMV, as the kids like to say.

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I don't think it would be as difficult as Jack puts it. Perhaps the standards are much higher where he teaches. Learning all the maths that is taught through secondary school is hard and requires discipline, but is doable in three months, especially if you have been exposed to the material before. In my experience, not much gets taught in the first three years anyway. It is more of a time to develop thinking skills, some of which you already have, since you're in a computing course. – Samuel Tan Oct 25 '11 at 10:40

I don't think there's a definitive answer to this, so just a few disjoint observations.

Fundamentally, all maths that has been written is designed to be understood by the reader, not to scare the reader. Try not to be intimidated by expressions like $F=1.8C+32$ --- this particular equation just tells you how to convert temperatures from Celcius to Fahrenheit; we could just say multiply the temperature in Celcius by 1.8 then add 32' but it's more efficient to write it as an equation. Other things you encounter in mathematics will be more complicated, but perhaps more than in any other subject, they can be broken down into simpler bits.

A string of equations should be followable by taking it line by line. You can follow it at your own speed. Treat yourself to a coffee (or a game of minesweeper or whatever) for each line in a long calculation or proof that you understand. When you understand something it's hard to forget it. Try to keep in mind what sort of problem you're trying to solve and why. (Why do we bother with integration? - To find areas. Why bother with the Sine rule? - To find unknown side lengths or angles in a triangle. Why do we need to find side lengths etc in a triangle? - It's essential in lots of contexts such as navigation and construction...) If you don't understand some proof or calculation, try to isolate what exactly you don't understand. Chances are there are only one or two things that are causing trouble.

From what you've said in your question, you're intelligent enough, committed enough and have the right attitude to succeed.

And I should say that MathStackExchange is a very useful resource -- use it for the problems you encounter when you've had a go at something and got stuck. We all need help sometimes. Good luck!

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A lot of us share the same feeling towards math. Maybe our high school teachers were terrible. Or maybe we also wasn't that interested in math at that point in time.

Time passed by and now you found yourself in a position that you really need to master the fundamentals of math in order to advance in your academic/professional life.

I'll share a secret with you: the only way I know that allows me to go and learn arithmetic, pre-algebra, algebra, geometry, trigonometry, probability and statistics, linear algebra and calculus` properly and in a fun way, is by going through the entertaining lessons offered by Khan Academy.

Warning: you will get addicted to math.

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Um, the OP (three years ago) already mentioned Khan Academy in his question ... – Henning Makholm Sep 17 '14 at 22:23
Thanks. Now he knows for sure it's the easiest way to go. – karlphillip Sep 17 '14 at 22:59