Programs What Do Math Majors Wish They Knew Before Graduation?

[espen180]

How about a more practical way of talking about this visual thing?

If you're a math major, you should at least know graphs of basic functions: conic sections, trigonometric functions, the square root function, logarithmic functions and be able to know what happens when you manipulate them. (Functions of one variable). What else?

Then you start getting into mutivariable. I can't draw these damn 3-d graphs. I can barely graph a point, (on ruddy 2 dimensional paper!). But I think I need to get some more facility of this, and familiarity with...the 3-d analogs of the above?

-Dave K

I am currently learning about manifolds, and since doing just about anything with manifolds has you climbing up and down chart functions (you will see expressions like D(\psi\circ f \circ \phi^{-1})(\phi(p))w_p ), and having a geometric image of what is going on is useful not to get lost in the equations. Although this is in an arbitrary number of dimensions, a 2-to-1 dimensional analog usually suffices.

 

[ingenvector]

Nope, it all relates back. It seems like LaTex is something "Math Majors should know" at some point or another. I know the physics department at our school uses it.

-DaveK

Well, that's good at least. My university's physics department doesn't seem to require LaTeX, though it is the norm and they do provide a preamble for submission formatting. Regardless, LaTeX should be something that everyone in a scientific, mathematical, or engineering field should learn, nearly every science or mathematics book and paper is published with it. I sometimes try and see if I can tell whether or not a publication was made using LaTeX. I suppose I'm somewhat of a typeset nerd.

Also, I think that using LyX, one can be nearly as fast typing notes in math as by hand, the only fall would be graphics would still need to be hand-drawn. What I sometimes do is type in some of my notes, the ones which are probably important and can use for future reference, into LaTeX, and it keeps a high degree of organisation, particularly if all the notes are in one comprehensive collection. I'm at around 250 pages now.

 

[ingenvector]

I'm way undergrad, so I think it's too early to know what it is I really "like." But I seem to be an "equation person," and somewhat outside of math I really enjoy learning about symbolic logic. But I think this is not exclusive of visual thinking. Whenever I glance at a topology book the first thing I see is some kind of set theory (which I haven't learned yet).

-DaveK

You do bring up a good point. Though I personally think it's largely useless, symbolic logic does not lend itself to visualisation readily unless the problem is already defined as a relational problem. It is also important to algorithms, though I don't think people usually visualise flow charts or trees beyond a certain size. What you are seeing in Set Theory I think is principally a similar syntax, but it doesn't use first-order or propositional logic and consigns itself to the more restrictive intuitionistic logic. It's what makes Set Theory so... intuitive.

 

[homeomorphic]

I'm way undergrad, so I think it's too early to know what it is I really "like." But I seem to be an "equation person," and somewhat outside of math I really enjoy learning about symbolic logic. But I think this is not exclusive of visual thinking. Whenever I glance at a topology book the first thing I see is some kind of set theory (which I haven't learned yet).

There are areas of math that are less visual than others. Modern math is all founded on set theory. Actually, set theory is pretty visual to me. A set is just a bunch of dots or sometimes a big blob in my mind's eye (or chalkboard or piece of paper). I can't remember things or reason effectively about sets if I don't visualize something. Even though it's abstract, point-set topology is pretty visual for me. I'm not just talking about topology. Analysis, topology, differential geometry, combinatorics, and even algebra (for example, there are lots of pictures you can draw of "root" systems in the theory of Lie algebras) all have their visual side. I think there's a place for less visual people in math, but there are large chunks of math that I don't see how anyone could make heads or tails of if they are geometrically challenged. Even if they can somehow manage, they'd be missing all the beauty of it. So, I don't think it's possible to be a broad or deep mathematician without being at least somewhat comfortable with visual arguments.

 

[Sankaku]

Hey, that's pretty cool. Now do you actually do your work in Latex or do you do it on paper and then re-type into Latex? Seems like I'd have a hard time working without physical paper and pencil. What editor specifically?

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 think there's a place for less visual people in math, but there are large chunks of math that I don't see how anyone could make heads or tails of if they are geometrically challenged. Even if they can somehow manage, they'd be missing all the beauty of it. So, I don't think it's possible to be a broad or deep mathematician without being at least somewhat comfortable with visual arguments.

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]

What editor specifically?

Vim!

 

[dcpo]

You do bring up a good point. Though I personally think it's largely useless, symbolic logic does not lend itself to visualisation readily unless the problem is already defined as a relational problem. It is also important to algorithms, though I don't think people usually visualise flow charts or trees beyond a certain size. What you are seeing in Set Theory I think is principally a similar syntax, but it doesn't use first-order or propositional logic and consigns itself to the more restrictive intuitionistic logic. It's what makes Set Theory so... intuitive.

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

 

[ingenvector]

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

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]

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

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]

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.

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

 

[dkotschessaa]

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.

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]

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.

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]

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

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]

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.

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]

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.
...

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]

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

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

 

[dcpo]

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

I'm a bit sceptical of this method as you describe it. I can't see how anything like the same amount of material could be covered, and the process doesn't sound very close to the practice of real maths, at least not as I have experienced it. Maths is either fully collaborative, with two or more people in a room working collectively on a problem, or independent, with most research involving a combination of the two. Real mathematicians will also use textbooks and papers to learn a subject, and only exceptional people can prove all the results contained in them themselves and still have time left over for creating original work. It's not so far from the standard model where students are asked to read things and then solve problems based on the material. The key difference is that real mathematicians have to work out the broader structure of the work themselves, formulating the problem clearly, deciding on relevant definitions etc. but the Moore method doesn't seem address this difference either.

 

[dkotschessaa]

I like the idea of this method, since I think pace is often substituted for understanding in most math classes. However I also agree with the above two posters. The compromise would be that I would like to see such classes offered as a supplement to regular classes - kind of like a "math lab" one credit course.

 

[Skrew]

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

This sounds horrible. I can't say I would ever want to be in a class like this.

 

[jk]

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

It starts out a little bit slow but the pace quickly picks up. You probably cover less material than a traditional class but you understand it much more deeply.

 

[Robert1986]

I have been enamoured with the Moore Method ever since I read Halmos' "Automathography". It seems like it would be very cool. Also, it sort of ties in very nicely with The Mathematician's Lament. (have you read this essay? its very good)

 

[jk]

I'm a bit sceptical of this method as you describe it.

That is a typical reaction from people who have not actually sat in a class (or taught a class) in this manner. But it has been used successfully in several places for decades now.

I can't see how anything like the same amount of material could be covered, and the process doesn't sound very close to the practice of real maths, at least not as I have experienced it. Maths is either fully collaborative, with two or more people in a room working collectively on a problem, or independent, with most research involving a combination of the two.

The process is actually VERY collaborative. When someone is on the board writing out their proof, the entire class is involved in helping them sharpen their argument by pointing out gaps, etc. It was one of the most participatory classes in math that I have ever seen.

Real mathematicians will also use textbooks and papers to learn a subject, and only exceptional people can prove all the results contained in them themselves and still have time left over for creating original work. It's not so far from the standard model where students are asked to read things and then solve problems based on the material. The key difference is that real mathematicians have to work out the broader structure of the work themselves, formulating the problem clearly, deciding on relevant definitions etc. but the Moore method doesn't seem address this difference either.

The typical math class involves assigned reading, lectures and homework. The first two tend to be somewhat passive activities because you are following someone's chain of reasoning. So you don't struggle as much to understand something. To me, an hour spent listening to someone lecture about the Heine-Borel theorem, for instance, is less useful than the same hour spent trying to prove it. You end up working much harder and understanding the topic more deeply, even if you don't succeed in proving the theorem, than you would have just following along in a lecture. It also stays with you longer.

 

[Robert1986]

I'm a bit sceptical of this method as you describe it. I can't see how anything like the same amount of material could be covered, and the process doesn't sound very close to the practice of real maths, at least not as I have experienced it. Maths is either fully collaborative, with two or more people in a room working collectively on a problem, or independent, with most research involving a combination of the two. Real mathematicians will also use textbooks and papers to learn a subject, and only exceptional people can prove all the results contained in them themselves and still have time left over for creating original work. It's not so far from the standard model where students are asked to read things and then solve problems based on the material. The key difference is that real mathematicians have to work out the broader structure of the work themselves, formulating the problem clearly, deciding on relevant definitions etc. but the Moore method doesn't seem address this difference either.

I haven't taken a Moore Method class, but (and someone correct me if I'm wrong) part of the idea (as I understand it) is that the instructor is there to sort of walk you through the stuff. So, its not a bunch of undergrads (or grads) in a room all re-discovering calculus. It is an instructor asking very well-placed questions and then usually forcing the students to work it out.

 

[jk]

I like the idea of this method, since I think pace is often substituted for understanding in most math classes. However I also agree with the above two posters. The compromise would be that I would like to see such classes offered as a supplement to regular classes - kind of like a "math lab" one credit course.

I think often times instructors try to cram as much information into a class as possible but that is counterproductive. I understand this if you're teaching a more practically oriented course like say, diff eq for engineers where it's plug and chug. But for courses intended for math majors, the goal should be to equip the student to be able to reason things out for him/her self and be able to follow the literature on her own. Learning has two sides, the presentation and absorption of knowledge. The lecture format accomplishes the first part very well but not necessarily the second, particularly when there is a lot of material.
I think combining the traditional approach with the Moore approach in the way you suggest would not work very well for two reasons: 1) It would be hard to split the material into parts that are proved by the instructor and other parts by the students. If the instructor is solving things for you in the "main" class, the motivation to do it yourself in the lab section is gone. 2) In a Moore Method class, you end up working much harder than in a regular class, at least in my experience. I don't know if a lot of students would have the time for both approaches in the same class.

 

[jk]

This sounds horrible. I can't say I would ever want to be in a class like this.

There are people who dislike this way. There were a couple of people in the class who just wanted things cut and dried: read section X of Chapter Y, do problems 3,4,5 and you're done. This way was too strange for them but I still think they benefited from the course.