Programs What Do Math Majors Wish They Knew Before Graduation?

[dkotschessaa]

Those of you in your upper level classes, graduate school, docs and post docs - what is it you wish you had known when you were an undergrad in mathematics? Things that nobody told you - but you wish they would have?

I have all sorts of silly questions sometimes on things ranging from what classes to take to what kind of pen and paper I should use sometimes. Perhaps you can help me and others out here by using the question above as a guideline.

-Dave K

 

[Robert1986]

Personally, I think that I did myself a disservice by not taking a topology class. I know some basic stuff from analysis, but not enough. So, I think that everyone should take topology (especially those who are going to grad school.)

 

[transphenomen]

If you can, you should audit at least one or two classes per semester. That way you can learn a lot more without it being too much of a time sink. You can do the homework of those classes when you have time or just ignore those classes when you need to prepare for tests in your real classes.

 

[nonequilibrium]

I mind not having had differential geometry in my undergrad, but the university didn't offer it is an undergrad course, and couldn't take a grad course in it, mostly due to there being too many course requirements already. Differential geometry seems really useful as I want to become a theoretical physicist and now I'll probably have to start learning general relativity without knowing differential geometry, which is presumably worse than studying quantum mechanics without knowing hilbert spaces, and I already deemed the latter as unacceptable...

EDIT:

If you can, you should audit at least one or two classes per semester.

Although that is a good idea, some people don't even have time for that.

 

[dkotschessaa]

I've thought about that transphenomen. I'm afraid I'm not too clear on my universities policy on auditing and it's been hard to find info on it. Does that usually cost something? Or can you just ask a professor if it's ok to sit in and do it unofficially?

Robert - I'm glad you said that. I was pretty sure I was going to take topology just out of interest, but it's seeming more indispensable now.

 

[dkotschessaa]

Does anybody wish they had taken a lighter course load and spent more time understanding the material? Or is the fast pace just "how it is" in undergrad years? (And then you can delve more deeply into your interests in grad school.)

I'm seriously thinking of limiting my course load, even if it takes me a year longer. I don't have any course requirements other than math right now.

 

[transphenomen]

I've thought about that transphenomen. I'm afraid I'm not too clear on my universities policy on auditing and it's been hard to find info on it. Does that usually cost something? Or can you just ask a professor if it's ok to sit in and do it unofficially?

At my college auditing is free. However, you obviously don't get a grade and you don't turn in homework or tests since they won't get graded. You just go there to sit for lectures. Finding information on auditing was very hard for me; there was only a vague paragraph about auditing in my 600+ page catalog. You best bet is to just email the teacher of the class you want to audit and ask if it is ok to audit his class. You don't even need the prerequisites to audit the class. For example, I will be auditing a graduate level general relativity class. When I emailed the teacher, he said he would assume the class knows Lagrangians and tensors. I don't know either, but I am studying up on them and find the probable challenge of the class exciting.

Also, even if you have time constraints, you can do your homework from other classes while auditing a class and just listen if something interesting happens.

 

[dkotschessaa]

Yes, I'm finding the same thing WRT to the search for information about auditing. It is a bit late in this semester anyway, but I'll consider next semester auditing linear algebra or something, which I'll take the semester after.

 

[transphenomen]

That's a good idea. There are two versions of real analysis and abstract algebra. One for applied and theoretical where the theoretical is harder. I am auditing the applied so I can ace the theoretical later.

Also, you can audit at anytime you want, even start in the middle of a semester. I did that with one class and I had a hard time catching up, but I learned enough to not need to try too hard when I take it for real later. For linear algebra, the first few weeks are just matrix multiplication, vectors, Gaussian elimination, and other things you should have known in a high school intermediate algebra class. If you start auditing it now, you should be able to keep up.

 

[moouers]

If you can, you should audit at least one or two classes per semester. That way you can learn a lot more without it being too much of a time sink. You can do the homework of those classes when you have time or just ignore those classes when you need to prepare for tests in your real classes.

I really wish I could do that. At my school, auditing a class costs the same as taking the class for full credit. It's silly.

 

[dkotschessaa]

I suspect it's the same at my university, but I'm double checking.

 

[Sankaku]

I'm seriously thinking of limiting my course load, even if it takes me a year longer.

If this is possible for you, I highly recommend doing it. Many people are constrained by financial considerations that make it difficult to take extra time for their degree. However, if you can take fewer classes you will learn the material much better (and get better grades as well).

 

[redrum419_7]

I really wish I could do that. At my school, auditing a class costs the same as taking the class for full credit. It's silly.

Same at my school

 

[dkotschessaa]

If this is possible for you, I highly recommend doing it. Many people are constrained by financial considerations that make it difficult to take extra time for their degree. However, if you can take fewer classes you will learn the material much better (and get better grades as well).

Yeah, whenever I mention it to people at my school they start talking about their scholarships and such, having to maintain a certain amount of credits, etc. I really don't have that issue. It will make things a bit more expensive as there are costs to every semester outside of just credits, but I'm ok with that.

 

[dkotschessaa]

Ok, here come some of the really silly questions I warned you about:

Paper and pencils.

Seriously - I can't seem to find out what my optimum tools are. I go through notebooks like crazy, and I can't seem to keep organized between class notes and homework. Now I'm even starting to wonder if I should just stop using notebooks and use *blank* paper, since this is what professors seem to use on overheads and this is how we are tested. Makes sense to do my homework the same way, right?

I haven't found the right writing utensil either. I keep getting mechanical pencils whose lead just falls out all the time. I end up having to carry a bunch of them around so I always have "backup."

Becoming more visual: Are there "equation people" and "geometric people?"

I tend to think I'm the former, but it seems that there's a lot of mathematics that depends on some kind of visualization, which I'm not that great at. Some teachers de-emphasize this, some teachers put a lot of emphasis on it. (Perhaps they themselves are in these two groups). Should I make a conscious effort to be more visual or just go with what I'm better at? I mean - knowing the shapes of different graphs, being able to work in three dimensional coordinate planes, etc.

-DaveK

 

[diligence]

Honestly, I think those are the type of thingsyou just need to figure out what works best for you. I don't see any general trends in my classes, some use pens, some use pencils, some use notepads, some use notebooks...I use blank paper and binders. The only thing I would recommend is either use a mechanical pencil or a pen so you don't have to sharpen the tip constantly.

As far as equation people vs geometric people, I think this is true to a degree. But again, you just need to do what works best for you.

 

[20Tauri]

Personally, I use recycled paper for my homework and take my notes in a notebook. My school collects computer paper that is printed on one side only to reuse in a recycle printer, so I just take paper from the collection bins and do my homework on the blank side. I like Pilot pens, but then I end up having to rewrite from the beginning if I screw up. My homework buddy turns in his work on gridded yellow paper and writes in mechanical pencil.

A topologist told me that he didn't really like algebra because it felt like symbolic manipulation to him and he couldn't visualize it very well, so I guess some people do prefer equations over geometry or vice versa.

 

[Sankaku]

Paper and pencils.

...snip...

Becoming more visual: Are there "equation people" and "geometric people?"

...snip...

I take notes on lined paper with a Pentel Kerry mechanical pencil using 0.5mm HB lead. Everyone has to find their own groove, though.

I submit any of my work in Latex. However, this is a time investment to learn properly so don't do it unless you are serious about carrying on in mathematics.

Strangely, I have a slight aversion to Point-Set topology because (to me) it seems like a mess of arbitrary unintuitive spaces with more counterexamples than anything resembling structure. Algebraic Topology is a bit of a different beast, though.

On the other hand Algebra is the most beautiful thing I have encountered in mathematics. I am deeply visual so I have found (admittedly strange) ways to visualize all the structures I have met in Algebra. Whatever way you build intuition is up to you. You will need intuition of some kind, though. If you rely on just cranking through equations there is a ceiling waiting for you...

 

[chiro]

I don't know if this is useful, but I would recommend that you know enough about other areas so that you understand why the other area is useful.

In other words know what a specific area is about in a few sentences but not the actual specifics of the proofs, identities, formulas and so on: just enough so that you know that if you have to figure out something out and you remember what a particular area is all about, then you go to the area later on and use what is already out there for your problem. It's also useful in advising other people who you work with directly or indirectly in your own field since the nature of mathematics is that it's connected and often a different perspective can end up solving a problem (which has happened countless times)

So if you a pure mathematician, know a bit about probability and statistics in terms of how to make a good inference. If you are a statistician, know about analysis in terms of convergence, continuity.

The thing is that this is becoming necessary anyway with number theorists using probabilistic primality testing and other examples of this.

 

[dkotschessaa]

I take notes on lined paper with a Pentel Kerry mechanical pencil using 0.5mm HB lead. Everyone has to find their own groove, though.

I submit any of my work in Latex. However, this is a time investment to learn properly so don't do it unless you are serious about carrying on in mathematics.

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?

Strangely, I have a slight aversion to Point-Set topology because (to me) it seems like a mess of arbitrary unintuitive spaces with more counterexamples than anything resembling structure. Algebraic Topology is a bit of a different beast, though.

On the other hand Algebra is the most beautiful thing I have encountered in mathematics. I am deeply visual so I have found (admittedly strange) ways to visualize all the structures I have met in Algebra. Whatever way you build intuition is up to you. You will need intuition of some kind, though. If you rely on just cranking through equations there is a ceiling waiting for you...

I feel sometimes that I am just plugging through formulas, however I'm only in Calc III. I find that it's hard to get a very deep understanding of anything given the pace of the classes. I feel that if and when I go and review the material on my own time I can get a more intuitive/visual grasp of it.

-DaveK

 

[ingenvector]

As an undergrad, I can't add to your main question other than reiterating what I heard from some acquaintances that numerical methods are often times lacking and, as someone had already mentioned, topology is important. Though, because of research in my prospective interests, I could also add geometry as a possible concern.

However, the question of pens and paper is of intimate and immediate concern to me. Though ostensibly subjective, I think that plain white paper makes for excellent work, however, one can easily be caught in an aesthetic trap sacrificing efficiency for aesthetics. This is why, from an academic perspective, lined-paper may be more advantageous. You can also try those large artist pads, writing on them well, filling the entire page with equations and graphs thoughtfully laid out, makes mathematics and aesthetics converge into one wondrous art of physical expression(s). I recommend mechanical pencils, I think the 0.7mm is a good and common range, and the Zebra f-301 pen.

As to "geometric vs. equation" people, often times professors understand certain problems in either extreme and want to press their knowledge of the subject from their perspective. I think that this is only human: to understand something particularly from one frame of mind and then to try and impart the perspective onto others to lead to the same conclusions.

 

[ingenvector]

I submit any of my work in Latex. However, this is a time investment to learn properly so don't do it unless you are serious about carrying on in mathematics.

One doesn't necessarily need to know LaTeX to profit from its benefits, what about LyX? It's a GUI for LaTeX so one doesn't have to learn its language. It's main insufficiency is its limited range of formatting though, but I would think if this was a concern then the point has already come where LaTeX should be learned anyways.

 

[dkotschessaa]

One doesn't necessarily need to know LaTeX to profit from its benefits, what about LyX? It's a GUI for LaTeX so one doesn't have to learn its language. It's main insufficiency is its limited range of formatting though, but I would think if this was a concern then the point has already come where LaTeX should be learned anyways.

I'll have to check that out...Thanks. Still wondering what editors people use to do raw latex though. I downloaded TexWorks awhile ago but it's been slow going.

 

[ingenvector]

LyX actually uses MiKTeX which is what I learned LaTeX in, though it isn't source like TeXworks would be. I quite like MiKTeX, though there doesn't seem to be any definitive standards, and it is so far sufficient. It seems that most editors were made by some group who needed basic LaTeX functionality plus some extra esoteric function that wasn't found before in the previous distributions, so really, in essence they are all basically the same. Actually, TeXworks would probably be quite representative of what you're describing, it may be that you will simply have to suffer through the experience.

But now I think I may have accidentally high-jacked the thread and brought it somewhat off topic...

 

[zif.]

I really wish I could do that. At my school, auditing a class costs the same as taking the class for full credit. It's silly.

Same at my uni, but that's because it's set up as an, "Oh ****! I ****ed up and don't want a 'withdraw' or, worse, 'not complete/fail' on my transcript" for students.

School's like this are playing pretty fast and loose with the term 'audit'.

That said, even if your school charges to 'audit' a class like mine does, there's not a single physics or math prof here that won't let you sit in on the classes (as long as you're not a distraction).

 

[bpatrick]

Here's a few of my thoughts on this thread so far:

auditing is great, I would highly recommend it. I've never had trouble auditing courses. One of my friends tried to audit an actuarial class and was made to pay for it because it was "university policy" to still register even if you were auditing, but I audited 2 pure math classes at the same university and never had to register nor pay. I think the profs, in general, have some sort of code where, if you're pursuing higher, pure math, then they welcome auditing, but if you want to audit stuff that you might benefit directly from (like free actuarial exam prep) or auditing finance courses (free CFA prep) is where you run into more trouble ... since so many people would do that just to pass various career exams. I've never run into any trouble auditing graduate level math at any university I've audited from.

As far as the paper/pencil/work stuff: I roll with a clipboard and use recycled paper (one side already printed) to do all my busy work / general problem solving. I also have a big white board at home with dry erase markers that are made from recycled stuff and can be recycled when they run out.

I don't really take notes, so I can't help you much there. I read ahead in the course so I can pay attention and let the material sink in a second time rather than mindlessly copying what the prof is saying/writing and not really absorbing it. I have some scrap paper handy if they do something interesting/new and I usually keep those around in a notebook until I have the material mastered, then just recycle it like all my used paper.

I found the Zebra M-301 0.5mm mechanical pencil is really nice for me. They are well built enough to not worry about them breaking but also inexpensive enough that I bought 3 packs of them (2 to a pack) about 10 years ago and don't get super worried that I'll lose them or whatever.

I always recommend taking as much linear algebra + algebra as possible, especially if you are thinking about grad school. You can never know enough linear algebra. That being said, even though topology is one of the "big three" that you'll probably take qualifying exams in, take as much analysis and algebra as you can while an undergrad.

If you want reading material that will help a little with some "what is grad school like" moments you might be having, look into: "A Mathematician's Survival Guide" by Steven G. Krantz.

 

[homeomorphic]

Becoming more visual: Are there "equation people" and "geometric people?"

I tend to think I'm the former, but it seems that there's a lot of mathematics that depends on some kind of visualization, which I'm not that great at. Some teachers de-emphasize this, some teachers put a lot of emphasis on it. (Perhaps they themselves are in these two groups). Should I make a conscious effort to be more visual or just go with what I'm better at? I mean - knowing the shapes of different graphs, being able to work in three dimensional coordinate planes, etc.

I should comment on this, since I'm Mr. Visual. Just having intuition is the more fundamental thing, rather than "visualization" or geometry. Most of all, I like to understand. I just find that visualization sometimes makes things totally make sense where they would otherwise be incomprehensible (I can follow the steps, but might not remember any of it, were it not for the pictures). It's also a great aid to memory.

The advantages of visualization seem pretty obvious to me because a lot of times, it took me a long period of not understanding it visually before I came up with a good visual explanation. So, I've done both in a lot of cases, and everything always makes so much more sense once I come up with the pictures. So, I suspect a lot of the people who are more on the equations side just don't know what they are missing.

I attribute my (relatively limited, but quite substantial, by layman's standards) success in math largely to the fact that I am good at visualization. It's the inspiration for a great many of my proofs.

Sometimes, I do wonder if maybe it's good not to convert everyone to my viewpoint because maybe we need people who actually LIKE ugly calculations. Then, I can leave all the dirty work to them.

In any case, visualization does improve with practice. When I read Visual Complex Analysis, I had to rehearse the arguments in my mind for a long time in some cases before I was able to see it clearly in my mind's eye. It doesn't necessarily come instantly. I've gotten a lot better at it, but even now, sometimes, it takes some thought.

 

[dkotschessaa]

LyX actually uses MiKTeX which is what I learned LaTeX in, though it isn't source like TeXworks would be. I quite like MiKTeX, though there doesn't seem to be any definitive standards, and it is so far sufficient. It seems that most editors were made by some group who needed basic LaTeX functionality plus some extra esoteric function that wasn't found before in the previous distributions, so really, in essence they are all basically the same. Actually, TeXworks would probably be quite representative of what you're describing, it may be that you will simply have to suffer through the experience.

But now I think I may have accidentally high-jacked the thread and brought it somewhat off topic...

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

 

[dkotschessaa]

I should comment on this, since I'm Mr. Visual. Just having intuition is the more fundamental thing, rather than "visualization" or geometry. Most of all, I like to understand. I just find that visualization sometimes makes things totally make sense where they would otherwise be incomprehensible (I can follow the steps, but might not remember any of it, were it not for the pictures). It's also a great aid to memory.

The advantages of visualization seem pretty obvious to me because a lot of times, it took me a long period of not understanding it visually before I came up with a good visual explanation. So, I've done both in a lot of cases, and everything always makes so much more sense once I come up with the pictures. So, I suspect a lot of the people who are more on the equations side just don't know what they are missing.

I attribute my (relatively limited, but quite substantial, by layman's standards) success in math largely to the fact that I am good at visualization. It's the inspiration for a great many of my proofs.

Sometimes, I do wonder if maybe it's good not to convert everyone to my viewpoint because maybe we need people who actually LIKE ugly calculations. Then, I can leave all the dirty work to them.

In any case, visualization does improve with practice. When I read Visual Complex Analysis, I had to rehearse the arguments in my mind for a long time in some cases before I was able to see it clearly in my mind's eye. It doesn't necessarily come instantly. I've gotten a lot better at it, but even now, sometimes, it takes some thought.

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

 

[dkotschessaa]

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

 

[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