How to become comfortable with mathematical notation?

March 18, 2009 1:11 PM

How did you learn to read and be comfortable with mathematical notation?

Since December I have been teaching myself math from pre-algebra to pre-calculus. I've been fine so far, using a variety of books and doing all the exercises. Now that I'm approaching more difficult topics like Calculus and Linear Algebra (not to mention Statistics and the Mackay book on inference), I am having a more and more difficult time with all the notation. Often I find the underlying ideas perfectly understandable, but the notation seems totally impenetrable, and it's really impeding my progress.

The question is, really: How did you learn to read and be comfortable with mathematical notation? Got any hints? Good books? Anecdotes?

posted by fake to Education (28 answers total)

Time and repetition. You've covered in a few months what most people are fed slowly over at least five years of plodding, group-oriented education. Slow yourself down. It's great that you're picking everything up so quickly, but it will be largely useless to you if you don't force yourself to really absorb the notation (there's a great little Feynman anecdote about this -- he taught himself some advanced topics and in the process created his own notation that no one else understood, and suffered as a result).

If you're really grasping concepts like statistical inference, then I have no doubt that infinite summation notation is within your grasp -- but you're going to have to just pound it into your brain. Be strict with yourself and be sure to always use proper notation when doing your own exercises, even if you feel it would be easier to just stick with what you know.
posted by telegraph at 1:23 PM on March 18, 2009

PS - Do you have a study group or anyone you can talk about your studies with? Talking about math is a great way to train yourself to comprehend notation and translate it on the fly.
posted by telegraph at 1:25 PM on March 18, 2009

Response by poster: I don't have any group options at the moment, but they might manifest in the future.
posted by fake at 1:27 PM on March 18, 2009

It's true that there is a lot of really horrible notation out there. Would you say that your biggest issue is with the notation itself or with how to go from "underlying ideas" to formally correct statements? My suggestion would be to try making up your own notation and to see where it leads you. You may realize there are subtleties in the underlying ideas that mean that it is tricky to define extremely simple notation for them. Or, you may come up with some notation that is genuinely simpler and better than what's standard in textbooks.

On preview, I see that not everyone agrees with my approach. The biggest thing is to do what works best for you to get a firm handle on the mathematics.
posted by louigi at 1:29 PM on March 18, 2009

Thinking about it, it seems to me that I learned to read mathematical notation somewhat the same way I learned to read English: by having someone read it to me first. I think watching someone write out a math problem on the board while they explain what they're doing ("Now we take the integral [draws integral sign] from t-nought to t [writes out limits of integral]...") is very helpful in learning what all the parts mean.

So perhaps you would find it helpful to watch videos of mathematics lectures on the subjects you're studying. MIT's OpenCourseWare has some videos of math lectures. ArsDigita University seems to have some as well, and I'm sure there are lots more out there. You might also find it helpful to hire a tutor just for one or two sessions to go over the notation.

Good luck!
posted by fermion at 1:42 PM on March 18, 2009

First, you've hit something deeper than you probably know: von Neumann said "In mathematics, you don't understand things. You just get used to them."

Probably the best way to get used to math notation is to read statements out loud:

1+1=2

One plus one is equal to two.

-b ± √ (b² - 4ac)) / 2a

The quantity negative b plus or minus the square root of b squared minus 4ac, divided by 2a.

Unfortunately, I don't know how to put limits, integrals and summation into meta markup, so the above may be insulting rather than enlightening.

You might also try making flash cards for the various symbols. Just the process of making them will help you give the symbols specific meaning, and if you find yourself unable to remember them without practice, you're prepared.
posted by pwnguin at 1:44 PM on March 18, 2009

This worked for me - your mileage may vary.

Learn where the symbols come from. For example the integral sign is just a funny "s" for sum. The d's in dy/dx for a differential comes from delta, meaning a small change. Read the equation out loud to yourself - instead of a frightening looking set of symbols it becomes e.g. "the definite integral from a to b of y with respect to x". Then break each bit down further until you get to stuff you understand.

And if you're still struggling with a particular equation, feel free to MeMail me and I'll see if I can explain it better.
posted by Electric Dragon at 1:47 PM on March 18, 2009 [2 favorites]

Some books are more notation heavy than others. Try more than one text. Personally I like really explicit notation until I'm very comfortable with a concept; but you might feel differently, plenty of people do.

It's great that you're doing all the problems; the best way to feel comfortable with the notation is to actually use it for computing things. Then you'll see why it's necessary to write down, say the "dx" in an integral (what if you were integrating it "dy" instead?). It can be tough to see why all the bits in notation are actually around until you have to use the information for something. (And it can be tough to deal when some bits are omitted until you realize they aren't always pertinent.)

You might also try upping the amount of problems you're doing; see if you can get a workbook/program/whatnot that gives extra practice.
posted by nat at 1:49 PM on March 18, 2009

Response by poster: Probably the best way to get used to math notation is to read statements out loud:

Thanks for saying this; it has worked very well for me so far. Maybe one of the limitations in self-teaching is the difficulty in verbalizing stuff like the integral sign? I mean, hard to speak it and even harder to grasp it.
posted by fake at 1:49 PM on March 18, 2009

Realize that mathematics is a language unto itself. It is perfectly normal to be at odds with the notation after only 3 months of study -- one semestre in a formal setting! It takes years to develop mathematical maturity and some things you just can't speed up,
posted by randomstriker at 1:55 PM on March 18, 2009

Try to teach it to your dog. Or your mom. Reading it out loud, as you've noticed, helps. Trying to pass the info on gets a completely different part of your brain. I didn't "get" a lot of upper calculus until I was requested to tutor it.
posted by notsnot at 1:58 PM on March 18, 2009

Response by poster: Realize that mathematics is a language unto itself.

Right, I don't mean to sound totally impatient, but I do have some goals and levels of understanding that I'd like to meet in the next two years, which is why I'm asking for help. :) I'm cognizant that some levels simply being unreachable in that span of time.
posted by fake at 2:01 PM on March 18, 2009

Thanks for saying this; it has worked very well for me so far. Maybe one of the limitations in self-teaching is the difficulty in verbalizing stuff like the integral sign?

It may be harder to figure out how the notation is verbalized from just reading stuff, but everything you see you should be able to read out loud.

For example, "the integral from a to b of the quantity x^2 +2 end quantity, with respect to x [this is dx] equals x^3 divided by 3 plus 2x plus a constant" should make perfect sense and be translatable back into the notation.
posted by leahwrenn at 2:10 PM on March 18, 2009

As to how I became familiar...well, I teach the stuff now, and as I'm lecturing, I'm speaking the notation at the same time as I'm writing it on the board, so I hope that my students are learning how to say the notation as well as to write it. But that doesn't help you any.
posted by leahwrenn at 2:11 PM on March 18, 2009

This is probably a very personal brainquirk and so I doubt will work for most, but anyways: I find it very helpful to think of formulas and notation as pictures in my brain. That is, for many math concepts I recall them easier if I think of how the formula/notation/definition/theorem parts looked like visually, shapes, position, density of the symbols, etc. Then I can typically "read" it textually from the picture in my brain, at least if it's not devilishly complicated and I have actually used it for some work during the last few years or so.
posted by Iosephus at 2:40 PM on March 18, 2009

Right, I don't mean to sound totally impatient, but I do have some goals and levels of understanding that I'd like to meet in the next two years, which is why I'm asking for help. :)

Care to share your goals? We might be able to offer better advice depending on what you're looking to achieve.
posted by telegraph at 2:40 PM on March 18, 2009

Not sure if this is true for all mathematics but in engineering a lot of greek letters were used in the mathematical expressions discussed.

I found it helpful to know what the (often) greek letters were called before I then started to read notation which made use of them - this allowed me to 'say' the expression in my head without have to resort to 'strange looking wiggly thing raised to power of x'.

There's a good resource on the greek alphabet in Wikipedia
posted by southof40 at 2:48 PM on March 18, 2009

i didn't read everything above so apologies if this is repeated: what i find really helpful is reading every notated line aloud or in my head in english. like "the integral of x is equal to the... etc." do this no matter how long it takes you the first several times or how often you have to flip around to find out the word for that notation. soon you'll be reading it like that in your head automatically and you'll know perfectly well what it means!
posted by skaye at 3:09 PM on March 18, 2009

If this is relevant, I'm attempting to learn from the MacKay book--I've been going at it for a few months, and I'm not sure it's a very good idea. The ideas are cool and sound, but it feels like there's this vast "understructure" and intuition he's left out for space and time reasons. He does use notation very, very, very flexibly (i.e. ambiguously), presumably because he can do all this stuff in his sleep and it's convenient shorthand.

So, fwiw, I'm very close to going back to more basic stuff before trying again, but I haven't found a good book yet. (M.S. in electrical engineering, but I was out of school for a while)
posted by zeek321 at 3:12 PM on March 18, 2009

Math isn't meant to be read; it's meant to be performed. If you go to a graduate level class, not uncommonly you will see someone at the front looking down at their notes, reading them aloud and copying them to the board. Everyone in the room watches, and copies. What the person up front is doing to making the written to verbal transmutation, and explaining a step, suggesting why they'd do the next step, etc.

If you couldn't do that, and be comfortable with why each is happening, you don't understand it. If you aren't doing associated problems, you don't understand it. You have to be able to interact with the concepts, to use them in not-explicitly suggested ways, to perform them, and explain them.

If you are somewhere and don't recognize the notation, you need to go back and find all the pieces. You really can't skim a math text.
posted by a robot made out of meat at 3:16 PM on March 18, 2009

Are you good with computers? Dabble a little (or a lot) in programming? Then you will have great fun with Mathematica. I found it to be the most useful tool by far for learning math. The student licenses are quite affordable.

The weakness of a book is that you can't interact with it, and there are a fixed set of examples and solutions provided. In a classroom environment, you have a professor or a TA to ask questions and get clarification. Not true for you when doing self-study.

Thus the real power of Mathematica for a student is to be able to manipulate algebraic symbols and see the results immediately, in front of your eyes. And also break down problems step by step, so that you can see how a solution was derived. And to come up with more variations of problems and solve them in real-time. It helped me a lot when I didn't have a prof I could ask questions of.
posted by randomstriker at 3:38 PM on March 18, 2009

And let me clarify -- you don't use Mathematica to cheat at your homework (you could, but that's missing the point). You use it to create more homework for yourself...you experiment with solving lots of problems on paper and then see if your answer matches what Mathematica spits out. Way better than waiting a week for a TA to hand back marked homework to you.
posted by randomstriker at 3:42 PM on March 18, 2009

Response by poster: I'll take a look at Mathematica -- my advisor uses it for his research.

Care to share your goals? We might be able to offer better advice depending on what you're looking to achieve.

Unfortunately they're only going to make me look more impatient. :) I want to have a good understanding of the Fourier transform and a number of EE/signals concepts. The idea is to be capable of entering a grad program at a place like MIT, where I want to work on image processing stuff. Think future cameras.

I have had the thought that even if I wasn't up to snuff on paper, at least I could use computer tools like Matlab and Mathematica to work around it. I'm currently studying image processing with Matlab, and using it (with Psychtoolbox) in my neuroscience research...

Zeek, thanks for the heads up on Mackay.
posted by fake at 3:50 PM on March 18, 2009

I think the key thing here is that your goal isn't actually to be comfortable with the notation, it is to understand the mathematics behind it. Being comfortable with the notation is necessary for understanding, but not sufficient. You have to get to the point where wading through the notation is no longer the barrier to writing down a correct statement.

I find writing things out several times to be more helpful with familiarizing myself with notation than speaking aloud. If there's a definition that I don't understand, I'll write it out, think about it as I do so, try to do an exercise related to it or prove a theorem about it, and if I can't then I throw the paper away, write the definition again, try another exercise, etc.

But you say that speaking aloud has helped you, so do what you will. In my experience, once I've written something down enough times, it sticks in my head.
posted by number9dream at 4:25 PM on March 18, 2009

I have had the thought that even if I wasn't up to snuff on paper, at least I could use computer tools like Matlab and Mathematica to work around it.

Er, no. You definitely need to understand the theory from first-principles -- the software is a tool, not a crutch.
posted by randomstriker at 5:50 PM on March 18, 2009

One thing that should be mentioned is that everyone is somewhat unhappy with math notation, especially as it's written out at elementary levels (you've probably noticed that the 'x' is no longer used in multiplication). Different fields use different notation for the same thing (sometimes out of stubornness, sometimes because of how it fits in with other math they need to use). For instance, dot products of vectors. Usually when taught to undergrads, it's written
a . b (hence "dot product", when the proper name is "inner product")
In physics, it's often written
< a | b > (bra ket notation)
or a_n b^n (Einstein summation notation: first n is subscript, the second is superscript)
I've seen it written as
(a, b)
as well. This is a trivial example, but each of those notations has profound impact on how it can be used in an equations, and how "a" and "b" should be thought about. For instance, the a_n b^n form generalizes nicely when the vectors live in a curved space. People got annoyed with existing notation and invented new notation to suit their purposes.

I suppose this is just a clunky way to say as others did, that you just eventually get used to the notation. Except sometimes it's such an impediment that you'll switch to a new, better notation in more advanced classes. Finally, it's simply the case that reading math is much harder than reading other things, and you have to go slower and really think about it. That's probably what happens when you read things out loud ---you slow down and focus. If you skim through equations, you're wasting your time and might as well be watching TV.

Your goals seem doable, especially if you can get some help.
posted by Humanzee at 7:13 PM on March 18, 2009

Response by poster: Thank you all. This thread was really helpful.
posted by fake at 8:36 AM on March 19, 2009

Maybe one of the limitations in self-teaching is the difficulty in verbalizing stuff like the integral sign? I mean, hard to speak it and even harder to grasp it.

The problem is that most math books don't explain how to pronounce this stuff. If you were taking a lecture course, you would have no difficulty with this as you would have heard it pronounced many times. It's a limitation of the book you are using, and I wish I had a recommendation to a good reference for how to say math notation out loud.

I have found being able to translate the notation into words is hands down the most helpful way to understand it.
posted by yohko at 1:30 PM on March 19, 2009