I thought that was very eloquently put, as far as it went. But it would help to know also: (1) what is your background in mathematics, and (2) why would you like to understand it better (besides doing well in courses). Do you expect you will need the mathematics you are studying now in your future career, or do you feel that some understanding of mathematics should be part of one’s general culture? What motivates you?

I’m guessing from what you’ve written that you may have had a sequence in calculus and perhaps a bit beyond that. I think your analogy to learning a foreign language is apt in many respects: mathematics is a precise, specialized language, with a complete grammar of its own and a rich, vibrant literature. The foreign language aspect becomes especially apparent when one first begins taking mathematics courses which emphasize an axiomatic development and mathematical proofs (and I think a good instructor in such an introductory course would be very sensitive to that foreign language aspect). But that’s not all: the cultural referents of the language (the structures of modern mathematics) may be something quite alien as well, e.g., the concept of a vector space.

There’s no royal road that I know of: learning mathematics requires a lot of patience and asking yourself or your peers or your teacher a lot of questions and thinking a lot on your own about what the hell some piece of mathematics is talking about (not just memorizing, but really getting underneath it). It’s pretty hard learning it by yourself; one piece of generic advice is to form a study group with similarly interested peers, or perhaps better, seek out someone who is a bit further along than you to talk to (it need not be a professor). This is no different from people who are learning Spanish getting together in informal discussion groups.

But it’s hard giving more specific advice without knowing a bit more about you.

Math is the most difficult subject I have ever taken. When I study for it, I actually don’t do well in my other courses in addition to not doing well in what I studied for. I often fall behind before I know it. It’s the only thing I can spend hours studying and still not understanding.

Math is abstract thinking? I consider myself good at abstract thinking. I can write poetry and analyze pictures and figures. Math is a foreign language? If math WERE greek to me, I would understand it better. I excell in learning foreign words and grammer. But math? It’s kind of like trying to analyze a picture that hasn’t been drawn or write and speak a language I don’t know.

It’s challenging, maybe even fun, but does it have to be like this? I would like to be able to take a math course for once in my life and remember the concepts, formulas, etc. for the duration I remember the other pertenant material I study.

How do people learn math when all they have is a serious of interrelated abstractions? For me, studying mathematics is mostly just memorization. Manipulating the pieces of the puzzle is difficult if one does not even know what the picture should look like, after all. I easily forget the concepts, and I find taking math courses very frustrating, because its a vicious circle that depleats my energy harms my other subjects, and usually leaves me feeling defeated, lazy, tired, and disappointed.

If there is any way to tame the study of mathematics, I would love to know, because I don’t want to be another semi-mathematically-illiterate professional in the American workforce. I want to understand and remember these concepts, but why do I have to learn how to speak a foreign language before I know what I’m saying?

I view this as kind of intellectually insulting. Why can’t I know what I’m learning? Don’t the teachers understand it? I wish I didn’t feel like I was in some kind of foreign country where no one was willing to teach me the language.

If someone would teach me the language, I too could become proficient in mathematics, but, as of now, I only know how to say a few words, and maybe ask for directions, and order food.

Another topic

Math should be taught as a complete foreign language with its own words, grammer rules, and pronunciation. Speaking a formula would certainly help one remember it better, at least it would me.

Why can’t someone wrap the concepts of math in the packages other curricula come in? For some people, learning math using the current methods is like trying to load a DVD into a CD ROM drive.

Well, that’s all I have to say.

Math is hard, just like learning any OTHER language is hard.

Sigfpe Says: Think of subjects like algebraic topology where much of
the fancy machinery is just a way to allow formal notation
to get a handle on things that a child already knows
intuitively.

You know a lot children with an intuitive grasp of homological algebra?!?
I mean, the 5 lemma sure, but I am sure most of them have a little trouble with the snake lemma.

Math is easy. Like many of my colleagues I sailed through my degree hardly lifting a finger to work. Once you grok a theorem, say, it becomes obvious and more or less unforgettable. I used to feel sorry for people studying a subject like chemistry who had to memorise an immense body of empirically determined facts to get anywhere. There’s little to memorise in mathematics and no need to work late into the night cramming arbitrary facts.

Math is hard. Just look at how much effort it takes to explain even the most elementary mathematics to a non-specialist. Pick up a mathematics textbook you’ve never read before. It’s not at all unusual for anything just one page beyond where you have already read to be completely incomprehensible. Mathematics texts pile up layer after layer of definition and abstraction which you have to ‘get’ before you can proceed. I can pick up a textbook in a subject like genetics, say, and get the gist of just about any chapter. Conceptually, he subject is more or less trivial and mastering it is just about learning some details.

Math is easy. Time and time again when you get to what appears to be a difficult result you realise it’s fancy language for a trivial intuition or concept. Think of subjects like algebraic topology where much of the fancy machinery is just a way to allow formal notation to get a handle on things that a child already knows intuitively.

Math is hard. I can’t tell you how many times I’ve been stuck there looking at a theorem knowing that everything I need is spelled out there in front of me and yet not being able to put the pieces together. It’s not like I can blame it on a missing fact that I need from another book that requires the work of going down to the library to look something up. This is a completely different kind of work - just you battling against your own mind. I don’t know of anything harder, sometimes, than trying to put those pieces together so that they fit together as a logical whole. You can’t just try to memorise the theorem. If you don’t get it then you won’t be able to use it. There’s no guarantee that if you just spend 12 hours looking at it you’ll have done the required work. No other subject begins to compare with the difficulty of doing this for a challenging proof.

Math is easy. Of course it is, I’m a lazy ba*tard and couldn’t have studied anything else.

Math is whatever you like.