Things math majors should know

Sankaku

I use Kile (KDE-Linux) but Texmaker seems to be a similar cross platform editor.
http://en.wikipedia.org/wiki/Texmaker

There are plenty of options out there. Just try them and see what you like.

I do very rough pencil work on paper before typing my work up. However, sometimes it is just as fast to work on the screen once you get used to the Latex notation.

Sankaku

I agree heartily with this. However, I am biased since I am very visual, too.

Dave, don't give up on visualizing curves & surfaces in calculus just because it is hard to start with. It is worth the time you put in.

Robert1986

Vim!!!

dcpo

Could you expand on this? I'm having trouble following what you're saying.

ingenvector

I would be glad to, but what exactly is it that you are having trouble following?

chiro

Interesting thread.

I think personally, at least from my limited experience with mathematics is that mathematics symbolically provides a kind of sensory language in its own right in comparison to the other sensory things associated with visualization (including things both statically and dynamically defined incorporating motion), and auditorial realization.

The thing is that mathematics is a way to build up sensory perception in ways that is hard to do with our normal visual and auditory sensory elements.

I do think it's possible to visually analyze things in higher dimensional spaces through a variety of projections, and in fact this technique is highly common especially in data mining and other similar endeavors.

But the thing about mathematics is that it is a compact representation: in other words, what the thing often represents (especially in highly abstract situations) is something that doesn't just relate to one thing but to something often with a high amount of variation captured in the expression.

This is what mathematics is all about: understanding variation whether its deterministic or non-deterministic variation: it's still just variation.

Considering that this variation is variation in every sense, such as the variation for quantities that are numbers, variation in the number of dimensions and properties of the space, variation in the operations used as found in groups, and generally variation in everything that even has the potential to have variation, it is not surprising the the more abstract representations in a compact form capture a humungous amount of variation in what is actually being described.

So what I see happening is that through the language of mathematics we are creating a kind of sensory input of its own that transcends what we are capable of using only the standard visual analysis that we are accustomed to intuitively than if we did not have mathematics or a context to put mathematics in and this is important because it will allow us not only to compress our understanding in a language that enables such compression, but to also juggle complexity and variation in a way that we never could before with just our eyes and ears alone.

ingenvector

Actually, I think I may know what you are asking after all. Let me backtrack since I was drunk with sleepiness when I wrote that post, as I am also now I suppose. But I digress...

In the context of visualisation, the most visual aspects of set theory are generally those which were first derived using intuitionist logic in naive set theory. Thus, the relations that the person was seeing in set theory that were visual were I think the intuitional syntactic structures of Set Theory which are also expressible in other logics. Naturally, nearly everything in mathematics is translatable, and first-order logic and the such is not exempt. I did not mean to imply that Set theory did not use first-order logic, indeed it can as in Zermelo–Fraenkel set theory, I merely meant that that was most likely not the visual component as this tends to delve into the equation classification over visualisation. I wrote rather sloppily, and I see now that it can be rather ambiguous and more open to interpretation than I had intended. I hope this answered your initial question.

dcpo

Thanks. What do you mean by 'intuitionistic logic'? Do you mean it in the sense of Brouwer and Heyting?

dkotschessaa

Yeah, I'm definitely going to spend some time on it.

There's some books out of "proofs without words" in my library that I think I'm also going to spend some time with this summer to improve my visual sense. It's not calculus but I think it may help.

dkotschessaa

Man, so I just discovered the joys of unlined paper...

mathwonk

Taking notes is maybe relevant or not depending on how the prof teaches. Some profs just come in and copy the same stuff onto the board that is in the book. In those classes there is no need to take notes at all. The right way to treat that situation is to read the book in advance, and make notes on things that are puzzling and then use class for repetition (it never hurts to hear something hard twice) and to ask questions.

In other classes the prof ignores the book, leaving it as auxiliary reading and presents a complete self contained course of his own devising on the board. If you don't take notes there you may not know what was in the course afterwards. All Harvard courses were like this in my day as an undergrad there. There was also a book that one was advised to read for an alternate version of the material.

In such a situation, one cannot fully benefit from taking notes unless one goes over them faithfully every night after class, filling details, thinking through the arguments and making remarks about things to raise in class or in office hours later.

A very few professors write out a pristine textbook version of the material on the board word for word, and expect you to memorize it and regurgitate it back on the exam. I had one such professor. Although I learned very little math from him, I did learn how to get a guaranteed A.

Many years later, in many of the courses I taught myself, in addition to having one or two books that covered most topics, I also wrote up a complete set of notes for every lecture and handed them out or made them available online. In that situation i cannot see much reason to take notes but many people still did so. Of course there are people who think that note taking aids in listening whereas I have the opposite experience, I can't write and think at the same time (as Lyndon Johnson would say I probably also can't walk and chew gum simultaneously).

Even in courses where the classroom material is an original version not replicated exactly in the book, some people advocate not taking notes at all but merely listening very closely. Then after class go to the library and write down what one recalls, i.e. taking notes from memory as it were.

This way one often gets a better feel for what is happening, and has a chance of asking better questions. It is scary though and most people would rather mindlessly write down notes they never look at afterwards, than just listen and get what they can at the time, even though that is often more.

I usually did not have the nerve not to take notes myself, and as a result, after a certain number of years I had a huge collection of pretty useless, even largely illegible, handwritten material, some on lined some on unlined paper, that I finally just threw out.

Sankaku

Yes, this is superb advice. In lower-level classes profs (usually) follow a book quite closely, but this often changes after the second or third year of undergrad. I have just had one of each of the two types of courses mathwonk describes, above.

For the one, I just wrote down the bits that the prof did differently than the book and listened for the rest.

The other, I wrote intensely for every second of the class. Now the course is over, I want to type up my notes in Latex. I haven't found a treatment of Ring Structure Theory in any textbook that is quite like what our prof gave us. It amazes me that every class he lectured for an hour and a half using only one page of his own notes... ...which he barley even looked at.

bpatrick

Haha. I almost included that in my previous post when I was talking about rolling with the clipboard + recycled (printed on one side + discarded) paper combo.

I haven't used lined paper since 2003 and have no intention of going back.

I concur with what mathwonk said above, and why I hardly ever take notes in class. It's so much better to read ahead so you know what is going to happen in lecture, then listen closely during the lecture rather than mindlessly copying stuff down. Keep some paper around for working examples along with the class or taking note of something that the prof may do that is super profound or gives a different take on the material / technique / whatever.

In case you missed it in my previous post:

A Mathematician's Survival Guide:
Graduate School and Early Career Development
Steven G. Krantz
Publication Year: 2003
ISBN-10: 0-8218-3455-X
ISBN-13: 978-0-8218-3455-8

well worth the read if you're seriously thinking about a career in math.

dkotschessaa

Definitely. Unfortunately my library doesn't seem to carry it - not even on ILL! But I'll keep an eye out. There's a lot of other great books to his name though that...speak to me. At the very least I'm going to start looking at some of his other stuff, but I"ll keep an eye out for this. I have quite a stack of unfinished books to finish.

-DaveK

dkotschessaa

I find that I do need to take notes, even if I don't understand (while taking them) what's going on, or else I disengage from the class. I have some pretty severe attention problems so it's the only thing that keeps me following along.

I've tried to read ahead, but I can never seem to fit it into my schedule. I manage for the first few weeks of class, and from then on it's just staying afloat.

Even though I'm not following at the time I'm taking the notes, something seems to get through to some other part of my brain. When I'm working out a problem I'll recall something from class and have the "aha" moment. I almost never review the notes unless I get stuck somewhere.

So far this seems to be working, though I'd prefer to do it the other way. (Book first). I'd like to say next semester I'll have "more time" as I'm only taking 2 maths and a German class, but I was just elected vice president of our math club. So there goes that extra time! But it'll be fun.

Dave K

jk

The best math class in my experience is a class where there was no book - in fact, we were forbidden to consult any book on the subject. The instructor did not lecture, and students took minimal notes. First day in class, instructor comes in, writes a few definitions and proves a couple of simple theorems. Then he writes a theorem on the board with no proof. Asks someone to prove it. The student then goes to the board and works out a proof. The rest of the class can not help him or her but can criticize the proof and find the gaps. Instructor only gets involved to clear up misconceptions, point out holes in the arguments by questioning the students in a socratic manner. If the student can't solve it he is invited to try during the next class and the class moves on to another problem. The first student now has "homework"

As the class progresses, the theorems and problems becomes harder and more substantial until we were proving pretty advanced stuff. Students really enjoy this as they are "creating" mathematics. You don't really need to take notes until the whole class has agreed on the solution that was presented on the board. Sometimes it takes two classes to solve something. You can spend hours working on that one thing in order to present it at the next class.

This way of teaching (called the Moore Method) is closer to the actual practice of mathematics. Once you have had a course based on this method, it is hard to go back to the traditional method where the instructor lectures for an hour and you just sit there and passively try understand it or are furiously scribbling notes so you can read them and understand later

espen180

But how much is covered? Wouldn't the pace suffer with this kind of method?