Other Should I Become a Mathematician?

[IGU]

qspeechc -

Jacobson's Basic Algebra I is available in a Dover edition. It's probably the level you're looking for and around $12 new at Amazon. Less dense than Lang, more extensive and a step up in depth from Herstein's Topics in Algebra.

It is somewhat dry, meaning you have to supply the enthusiasm.

-IGU-

 

[mathwonk]

qspeechc: To put it another way, recall the famous quote: 'when asked how he had managed to make such progress in mathematics despite his youth, Abel responded, “By studying the masters, not their pupils.” '

 

[dkotschessaa]

Well unmotivated because they seem to just come out of nowhere. The book I'm using says do this and this and you will get this. But I don't blame it, deriving it seems tricky-- you need a bunch of clever manipulations that aren't so straightforward.

This might make you hate them more, but whenever I get stuck on something like this I always like to know something of the history of it. The first known use of partial fraction decomposition was by Isaac Barrows, in his proof of the integral of the secant function: http://en.wikipedia.org/wiki/Partial_fractions_in_integration

Following on the advice of Mathwonk to make your own exercises, (way early in this thread) this is another place where this is useful. Take something like [5/(x+5)] * [8/(x^2+2)] or something like that. Multiply it all together, then try to decompose it again. Maybe integrate it before and again afterwards to show yourself how everything fits together. Then make more complicated problems.

If you're a real math geek this will actually start to become enjoyable...

-DaveK

 

[homeomorphic]

If you're a real math geek this will actually start to become enjoyable...

Perhaps, but I wouldn't want anyone to get the impression that you have to like that sort of thing to do math. It's much more interesting than that, thankfully. I'm sure there's a place in math for those who are thrilled by things like partial fractions. But there's a place for those who are not thrilled by them.

Partial fractions? Just learn them so you can get a good grade and be better at integration and then move on to better things. It would be much more interesting to design some Turing machines or figure out how to do some ruler and compass constructions. Something that has some intellectual content to it.

 

[dkotschessaa]

Homeomorphic, you are correct. That pretty much just came out wrong.

 

[mathwonk]

I agree with dkotschessaa that partial fractions is just a way of reversing adding fractions. it may seem more natural when you study complex analysis and poles and laurent expansions.

as a general rule, there is nothing at all that has no value and no interest, it is just being taught that way. I have a friend who is really really smart, and every time i say to him that something is rather boring or uninteresting, he ALWAYS says back: well what about this?... and it becomes fascinating...

 

[dkotschessaa]

I think one issue is that all these topics get thrown into textbooks and kind of whiz by kind of quickly (this is just the nature of the study I suppose) when really we don't get the story behind them. The truth is for every section of your calculus book there was likely a mathematician or two or more who spent serious time coming up with that particular technique or mathematical idea. There are people behind those ideas. This emphasis I find lacking. Maybe it's just me.

-DaveK

 

[Locrian]

Any ideas about investment banking?

Not really my thing. I can tell you that there isn't a single answer to the question you asked. What education you'll require will instead depend on what you want to do at the investment bank. Trader? Quant? Systems? Janitor? Different requirements.

 

[Nano-Passion]

Thanks everyone. :)

I think one issue is that all these topics get thrown into textbooks and kind of whiz by kind of quickly (this is just the nature of the study I suppose) when really we don't get the story behind them. The truth is for every section of your calculus book there was likely a mathematician or two or more who spent serious time coming up with that particular technique or mathematical idea. There are people behind those ideas. This emphasis I find lacking. Maybe it's just me.

-DaveK

That is one of the things that really irk me in our education system. The history gives so much motivation and context.

 

[dkotschessaa]

It's a state of affairs that isn't acceptable in the humanities but for some reason it is in the sciences. You just have to take it up on your own.

 

[qspeechc]

qspeechc -

Jacobson's Basic Algebra I is available in a Dover edition. It's probably the level you're looking for and around $12 new at Amazon. Less dense than Lang, more extensive and a step up in depth from Herstein's Topics in Algebra.

It is somewhat dry, meaning you have to supply the enthusiasm.

-IGU-

Great suggestion, seems perfect, thanks

qspeechc: To put it another way, recall the famous quote: 'when asked how he had managed to make such progress in mathematics despite his youth, Abel responded, “By studying the masters, not their pupils.” '

Yes, a good education lasts all your life, so I suppose I will get Artin's book, and Jacobson's too. And then Lang. Whew, mathematics is a slog! (But a good slog!)

 

[Nano-Passion]

It's a state of affairs that isn't acceptable in the humanities but for some reason it is in the sciences. You just have to take it up on your own.

You mean it is a state of affair (including a bit of history and context into the curriculum) is acceptable in the humanities but for some reason it isn't in the sciences?

 

[Mépris]

I still find it somewhat distressing that there are so many gaps in my pre-calculus background, which is something I expressed here a few months back. I have other commitments, so have not been able to fully immerse myself in that. The truth though, is that I don't particularly find learning (high school) algebra interesting. Sure, there are some parts I am intrigued about but the very thought of many of the things, such as being able to show that "if p and p + 2 with p ≥ 5, are both primes then the number p + 1 is always divisible by 6", leaves me unmotivated and a to a greater extent, frustrated. Looking forward, all I see is a series of hoops called "Pre-calculus", "Single variable calculus", "multi-variable calculus", "Linear algebra", etc...

I understand there are few things with my algebra that do need taken care of, but I figure I can take care of those loose ends as I move forward. It feels more tedious than actually fun. Perhaps it's the "collecting and not reading books syndrome", where people feel guilty about not reading books they were supposed and end up just collecting them. At any rate, I find probability and differential equations (very, very basic stuff, such as simple y = kx models for increase/decrease in number of bacteria or fish, but I know there's more cool things to be done with those) quite interesting - not to mention fun - and would rather learn those more thoroughly, so that I can start learning from a proper intro physics text and perhaps get into some more applied math.

I also recalled a post you (mathwonk) made, where you said that studying from a book should be done with the aim of learning something from it, not necessarily reading it line by line. (I'm paraphrasing here...) Can that be applied here? I really would rather just get ahead but, if that wasn't obvious enough, I'm at a loss here.

Another question. I noticed that MATH 25 and 55 at Harvard, Rudin's text is used. http://www.math.harvard.edu/pamphlets/freshmenguide.html seems to suggest that for MATH 25, the students could do just fine even if they've had a rather limited exposure to both subjects. Is this not somewhat premature? I thought learning from Rudin's book was usually after one had studied proof-based calculus courses, say both volumes of Apostol.

 

[Nano-Passion]

At any rate, I find probability and differential equations (very, very basic stuff, such as simple y = kx models for increase/decrease in number of bacteria or fish, but I know there's more cool things to be done with those) quite interesting - not to mention fun - and would rather learn those more thoroughly, so that I can start learning from a proper intro physics text and perhaps get into some more applied math.

I find that studying differential equations is much more fun than diving straight into learning how to find every derivative and integral of elementary functions. I feel very unmotivated when I finish one section of integration and the next one is just the same, akin to "you've learned how to integrate this type of function, now learn how to integrate that type of function." It is too computation based--something that I can leave for computers to take care of.

I find it much more interesting to study differential equations, and referring back and forth when required to differentiate or integrate x, y, or z function. And in fact, you feel much more motivated once you are in that context. Is everyone under the belief that learning everything in a linear manner is the best way? Because I certainly don't. And I find that it takes all the fun out of everything. I hope more people will realize it and things change.

 

[mathwonk]

In past years, math 55 at harvard has used a variety of books, all aimed at someone who has already had preparation comparable to apostol. they have used apostol's mathematical analysis, dieudonne's foundations of modern analysis, flemings functions of several variables, loomis and sternberg's advanced calculus, and notes by Wilfried Schmid. All of those books are more high powered and I think better than Rudin.

Most upper level courses at harvard can be very very advanced, and math 55 is one of the most ridiculously hard courses in the country.Sure read whatever interests you and use that as motivation to go back and learn more elementary stuff when you need it.

 

[qspeechc]

This might be interesting for undegrads looking to go to grad school. It's the topics Columbia grad school expects all entering students to know:
http://www.math.columbia.edu/programs/main/graduate/gradknowledge.html

 

[mathwonk]

I know this is absurdly hard to do, but still I assure you it is worth it, to try consistently just to understand a small amount of mathematics, i.e. one idea at a time, really well. Do not make it a goal to read a whole book. That is ok, but the point is to learn one idea at a time. I speak from experience.

 

[SophusLies]

I know this is absurdly hard to do, but still I assure you it is worth it, to try consistently just to understand a small amount of mathematics, i.e. one idea at a time, really well. Do not make it a goal to read a whole book. That is ok, but the point is to learn one idea at a time. I speak from experience.

This is excellent advice. When I was an undergrad I bit off way more than I could chew multiple times. There were a couple of quarters that I took 8-9 math/physics courses. I survived but at the time I thought I was learning more because I was covering all the bases. This couldn't have been farther from the truth because I was just learning everything in a trivial way.

Now in grad school I'm almost done with all my classes. Grad classes are much harder but you don't take as many each term and the ideas are fully developed. I feel I'm learning more than I ever thought possible recently because I can focus entirely on a smaller amount of material than a surface scratch of a whole bunch of subjects.

 

[dkotschessaa]

I know this is absurdly hard to do, but still I assure you it is worth it, to try consistently just to understand a small amount of mathematics, i.e. one idea at a time, really well. Do not make it a goal to read a whole book. That is ok, but the point is to learn one idea at a time. I speak from experience.

That's what I try to do in my own time, but unfortunately undergraduate classes are structured to give you as many different topics as possible in the shortest possible time. I feel like I'm always saying "Wow, that's a really cool idea...wish I had time to understand it."

 

[homeomorphic]

That's what I try to do in my own time, but unfortunately undergraduate classes are structured to give you as many different topics as possible in the shortest possible time. I feel like I'm always saying "Wow, that's a really cool idea...wish I had time to understand it."

Unfortunately, so are many graduate level classes, so I felt the same way in grad school. Good thing I'm done with classes.

 

[mathwonk]

then try it after the class is over. take just one theorem from the class and really try to understand it. eventually you will have a few key ideas that you really understand, and everything else will seem like a simple corollary of those. e.g. after decades of teaching studying and writing about it, I can say that all of the structure theory of an advanced linear algebra class, jordan form, rational canonical form, and so on, is a simple consequence of the euclidean algorithm. So if you want to understand the structure of finitely generated modules over Euclidean domains and then pid's, first learn well the euclidean algorithm. then see if you can understand why this is all there is at work in those other theories.

for non commutative algebra, a basic idea is a group acting on a set.

for commutative ring theory, a fundamental result seems to be the noether normalization lemma.

in manifold theory, the basic theorem is the inverse function theorem, and then the implicit function theorem. In many situations, a key result is green's theorem, and then its generalizations, the general stokes theorem.

 

[mathwonk]

On the theory that you can do more math if you live longer i feel this link is relevant.

http://www.nytimes.com/2012/04/15/h...tied-good-habits-to-longevity-dies-at-97.html

Basically a public health professor proved long ago statistically that you live about 10 years longer if you:

"do not smoke; drink in moderation; sleep seven to eight hours; exercise at least moderately; eat regular meals; maintain a moderate weight; eat breakfast.”Now that's not so hard.

Moreover a person who does fewer than three of these is only as healthy at 30, as someone who does at least 6 of them is at 60.

 

[Cod]

Does anyone have any topic ideas for a numerical analysis project? Instead of a final exam in my numerical analysis course, my instructor wants us to put together a formal proposal and provide a written professional report or paper. Other than that, my instructor provided the following guidance:

You proposal should include a description of the problem and your approach to solving it. For example a project could be to construct a general text classifier and the approach to the solution could be Bayesian statistics or least squares with the text turned into vectors.

I'd like to do something as it applies to computer science (my major) or baseball (my hobby). I've found a few papers online dealing with subjects of interests, but mostly at a graduate and beyond level. Any ideas, guidance, etc. is greatly appreciated.

 

[dkotschessaa]

And get married: http://wellbeingwire.meyouhealth.com/physical-health/married-men-live-longer-than-bachelors-study-says/

Having said that, it's 9:30, and my wife and I are going to bed early, so we can get up and have breakfast, for I have a Calc III test...

Good night!

-DaveK

 

[Plaristotle]

How important is it for math majors to have experience with programming? I'm able to take only one more first-year course and I'm considering taking an introductory course on programming (using Python). Specifically I'm interested in pure mathematics and I'd eventually like to go to grad school, if that helps. Any advice?

 

[chiro]

How important is it for math majors to have experience with programming? I'm able to take only one more first-year course and I'm considering taking an introductory course on programming (using Python). Specifically I'm interested in pure mathematics and I'd eventually like to go to grad school, if that helps. Any advice?

You should have done an introductory subject by the end of your undergraduate and along the way you will probably do stuff in something like MATLAB, Maple, R, SAS, Excel, or something else.

Even with writing papers it can be handy to know simple constructs if you have to do fancy stuff for generating figures and so on.

One other thing is that if you want to test statistical theorems, then it's a really good idea to have some programming experience to test your ideas before going the whole nine yards and proving something. This happens a lot in mathematics and unfortunately lots of people don't see the entire plot, but the climax so to speak.

Also one thing to remember is that if you can't get a job in pure math but there is stuff available that is applied where you work on a computer, produce models, run simulations and write reports or give advice, then this is a good thing to have under your belt in comparison to if you had no idea what a for loop is.

 

[dkotschessaa]

Does anyone have any topic ideas for a numerical analysis project? Instead of a final exam in my numerical analysis course, my instructor wants us to put together a formal proposal and provide a written professional report or paper. Other than that, my instructor provided the following guidance:

You proposal should include a description of the problem and your approach to solving it. For example a project could be to construct a general text classifier and the approach to the solution could be Bayesian statistics or least squares with the text turned into vectors.

I'd like to do something as it applies to computer science (my major) or baseball (my hobby). I've found a few papers online dealing with subjects of interests, but mostly at a graduate and beyond level. Any ideas, guidance, etc. is greatly appreciated.

Merely expressing my jealousy. I wish we had this option!

 

[Mépris]

then try it after the class is over. take just one theorem from the class and really try to understand it. eventually you will have a few key ideas that you really understand, and everything else will seem like a simple corollary of those. e.g. after decades of teaching studying and writing about it, I can say that all of the structure theory of an advanced linear algebra class, jordan form, rational canonical form, and so on, is a simple consequence of the euclidean algorithm. So if you want to understand the structure of finitely generated modules over Euclidean domains and then pid's, first learn well the euclidean algorithm. then see if you can understand why this is all there is at work in those other theories.

for non commutative algebra, a basic idea is a group acting on a set.

for commutative ring theory, a fundamental result seems to be the noether normalization lemma.

in manifold theory, the basic theorem is the inverse function theorem, and then the implicit function theorem. In many situations, a key result is green's theorem, and then its generalizations, the general stokes theorem.

I'll keep this somewhere where I can find it, for when I study linear algebra next year. I think you laid out differential equations in the same manner in another post. I will try to come to a conclusion on my own first, though.

This might be my favourite thread in this forum. It's the kind of thing that would have benefited me greatly back in high school when I started writing. We don't have very good libraries here, which made finding books a little harder. I don't think there's a very rigid sequence of books that one should read or study but there are some essential things that one should do in philosophy, history and politics and literature, if they are interested in writing. At any rate, all this is to say that I've observed that guidance, if available in high schools (I've been to three!), is usually poor, from someone who just does not care. That's why I like this thread.

Plaritotle, instead of paying for such a course, you could learn programming using Python in your free time from OCW Scholar. It's a clearly laid out course, so you shouldn't be encountering too many bumps. My intention is to do a little of this every day as from June.
(link

 

[Plaristotle]

Ah, that's some good advice, chiro. I haven't taken any yet, but I will eventually be taking at least a few courses in which R or MATLAB will be used.

Mepris, I'll definitely consider learning Python using OCW. Thanks for the link!

 

[noobilly]

I'm wondering about abstract algebra. I was doing Calc 1-3 and although I had to put in plenty of effort, it still made sense to me. But now when I look at some notes on abstract algebra it doesn't seem to be comprehensible to me at all. Am I missing some kind of prerequisite here, or have to plug some kind of hole in my mental process?

 

[homeomorphic]

I'm wondering about abstract algebra. I was doing Calc 1-3 and although I had to put in plenty of effort, it still made sense to me. But now when I look at some notes on abstract algebra it doesn't seem to be comprehensible to me at all. Am I missing some kind of prerequisite here, or have to plug some kind of hole in my mental process?

It might help to get comfortable with proofs, elsewhere. You can try naive set theory, first, or something like that. You have to get used to the abstraction of modern math.

Also, probably most math books/notes are overly formal/unmotivated/boring, so they don't convey how to think about the subject very well, especially for a beginner.

I have heard Pinter's abstract algebra book is pretty well-motivated. Another interesting one is Nathan Carter's Visual Group Theory. Symmetry by Hermann Weyl is another. Once you have groups down, you can try Galois Theory, by Ian Stewart (he starts with subfields and subrings of the complex numbers which are a very good motivating example for the general case and arise naturally in the context of Galois theory). The more formal books have their merits, despite being very inadequate in some respects.

 

[noobilly]

Thanks, I think you hit the nail there - I'm not very great at grasping abstract concepts. For calculus i usually have to work through a few concrete examples to get the mechanics down, then think of an intuitive explanation, before I can look back at the proof and really "get it". So this learning style won't work as I move up in math?

 

[homeomorphic]

Thanks, I think you hit the nail there - I'm not very great at grasping abstract concepts. For calculus i usually have to work through a few concrete examples to get the mechanics down, then think of an intuitive explanation, before I can look back at the proof and really "get it". So this learning style won't work as I move up in math?

Maybe something like that could work sometimes, but it will need adjustment. Its different. The end goal in calculus is usually to calculate things. With abstract algebra, the end goal is to prove things. So, you can't start with calculations.

I never really learned that way, so it's hard for me to say how it could be adjusted. I always wanted to understand everything first, and then use it, rather than the other way around. In undergrad, I usually just liked to read and convert all the proofs into pictures that I could visualize (or sometimes just moving symbols around in my mind's eye for algebra--but there are many pictures that help, too, in algebra). If the proof was too tough, I would realize it halfway through this process, give up and just try to understand the statement and how to use it, postponing an understanding of why it was true. Then once I made everything obvious enough for myself, I would tackle the problems. I did pretty well, but I don't know if that's the "right" way to do it or not.

A principle that I came up with long ago was this: If you think about anything long enough, it will eventually make sense. So, that's what I did. Just think until it made sense. Every time. In more advanced math, sometimes, you have to think a long time before it clicks.

 

[Group_Complex]

homeomorphic, did you take differential geometry as an Undergraduate? If so, can you recommend some good texts?

 

[homeomorphic]

I guess I was a 1st year grad student when I took differential geometry. I don't know that I would recommend the book we used. I'm not sure which book I would recommend for that. Visual Complex Analysis has a good section on it (actually, the author, Tristan Needham is rumored to be working on a differential geometry book that I'm sure will be mind-blowing), but it doesn't get into much detail. Also, there's a very similar discussion in Geometry and the Imagination. Probably do Carmo's book is okay for curves and surfaces, but I don't like his Riemannian Geometry book, which is sort of the sequel.

Some of these things, I just sort of learned from sources all over the place, and I never really bothered to track down the best book out there. Too little time.

 

[Group_Complex]

I guess I was a 1st year grad student when I took differential geometry. I don't know that I would recommend the book we used. I'm not sure which book I would recommend for that. Visual Complex Analysis has a good section on it (actually, the author, Tristan Needham is rumored to be working on a differential geometry book that I'm sure will be mind-blowing), but it doesn't get into much detail. Also, there's a very similar discussion in Geometry and the Imagination. Probably do Carmo's book is okay for curves and surfaces, but I don't like his Riemannian Geometry book, which is sort of the sequel.

Some of these things, I just sort of learned from sources all over the place, and I never really bothered to track down the best book out there. Too little time.

Why did you end up going into topology rather than differential geometry? It seems from my own humble studies of these subjects that whilst there is visualisation involved in Topology, it seems to be of a higher degree in Differential Geometry.

 

[dkotschessaa]

then try it after the class is over. take just one theorem from the class and really try to understand it. eventually you will have a few key ideas that you really understand, and everything else will seem like a simple corollary of those. e.g. after decades of teaching studying and writing about it, I can say that all of the structure theory of an advanced linear algebra class, jordan form, rational canonical form, and so on, is a simple consequence of the euclidean algorithm. So if you want to understand the structure of finitely generated modules over Euclidean domains and then pid's, first learn well the euclidean algorithm. then see if you can understand why this is all there is at work in those other theories.

for non commutative algebra, a basic idea is a group acting on a set.

for commutative ring theory, a fundamental result seems to be the noether normalization lemma.

in manifold theory, the basic theorem is the inverse function theorem, and then the implicit function theorem. In many situations, a key result is green's theorem, and then its generalizations, the general stokes theorem.

This has stuck in my head since you posted it. (You always do that, mathwonk!)

One particular sticking point in the calculus sequence was a large chapter on series and sequences, divergence, convergence, etc. It was probably one of the most difficult subjects. But it also seems to be where some of the really interesting mathematics is, and where you can study cantor sets and such.

I was thinking of spending a good deal of time on my own reviewing and researching in this area, maybe even putting together a guide for undergraduates that I want to bring to the tutoring center next semester.

I was thinking along the same lines. If I understand that topic very well I will understand calculus in general much better - much of which can be understood in terms of riemann sums (even the definition of an in integral, which we cover but don't really explore).

Do you think this would be a good area to delve into WRT to what you just said above? I'll be done with the main calculus sequence this semester.

-Dave K

edited to add: Perhaps the ultimate aim would be to really understand Taylor/Maclaurin series.

 

[homeomorphic]

Why did you end up going into topology rather than differential geometry? It seems from my own humble studies of these subjects that whilst there is visualisation involved in Topology, it seems to be of a higher degree in Differential Geometry.

Actually, topology seems more visual to me. I do geometric topology, so topology of manifolds. Specifically, low-dimensional manifolds, in my case, and more on the visual side of it. It's not just a question of topology or geometry--some topologists are more visual than others, and the same goes for geometry. A lot of geometers are really into ugly calculations. You can't judge from what it's like just starting out in the subject.

I do enjoy hyperbolic 3-manifolds, though, which involves a bit of geometry. And by the way, Thurston's book on that subject is a good place to get started on geometry, once you have the prerequisites for it.

 

[mathwonk]

dk that sounds great. convergence is fascinating. here's a little problem that got my attention when i taught honors calc; suppose you have a riemann integrable function f on the interval [a,b], and you integrate it from a to x, to get an other function g. how do you characterize g? well if f is continuous, then g is the unique differentiable function with derivative equal to f and with g(a) = 0.

But what if f is only integrable? Then it turns out that actually f must be continuous except on a set of measure zero, and that g is differentiable with derivative equal to f wherever f was continuous. Moreover g is not just continuous but Lipschitz continuous on [a,b]. I fact g is the only Lipschitz continuous function on [a,b] with g(a) = 0, and with g'(x) = f(x) at every x where f is continuous.

But it is NOT enough to just assume g is continuous with those other properties, you have to assume the stronger Lipschitz continuity.

I.e. there can exist a continuous g, with derivative equal to f at every x where f has a derivative, and g(a) = 0, and yet g is not the integral of f. Such examples are constructed using Cantor functions.

I.e. we can have a Cantor function g that is continuous everywhere and differentiable with g'(x) = 0 except on a closed set S of measure zero, and g(a) = 0, and yet g(b) = 1, Then g is not the integral of the function f which = 0 except on S where it equals 1, since that integral is identically zero.

The point is: how do you generalize the mean value theorem to cover a function g that only has derivative zero off a set of measure zero? How much more do you need to assume so that g is a constant? Ordinary continuity won't do.

 

[dkotschessaa]

I'll have to look into that mathwonk. It seems that I'm right about this being a central topic with a lot of fascinating ideas. Of course this is also where we get introduced to Euler's formula and some derivations of pi. Lots of great stuff there.

Final stretch. This coming week is my last week of the semester before finals. Though at this point I feel I can no longer measure in "how much time" (which is not much) but in terms of "how much stuff do I have to do/read/study." (Which is a lot).

-DaveK

 

[battousai]

http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra

If i successfully complete the course above, will I be ready for graduate level algebra?

 

[Group_Complex]

http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra

If i successfully complete the course above, will I be ready for graduate level algebra?

You must also be able to do most of the problems from a good text on abstract algebra. Simply watching videos will not be enough.

 

[battousai]

You must also be able to do most of the problems from a good text on abstract algebra. Simply watching videos will not be enough.

of course. that's what i meant by "successfully complete".

 

[dkotschessaa]

Since I'm still in undergraduate until at least Fall 2014 and not getting any younger, I'm starting to contemplate whether I should apply directly for a Phd. program.

Disadvantages: The time commitment. It's not over until it's over, or until (as I just read an old thread here, mathwonk says "until the fat guy says it is.") It'll also be an extraordinary jump in the kind of stuff I'm doing - but I think I'm mature enough to make that jump.

However if after 4 years, for some reason I can't finish, I won't have much to show for it.

Advantages: may save some time over FIRST doing a masters THEN a Phd. It might actually be funded rather than me having to pay (if I understand correctly), even if not very much, which is still more than what I get now, which is "zero minus tuition."

What am I missing, or where am I wrong?

 

[mathwonk]

the only thing wrong with going to school is burnout. (except for poverty). So if you are well motivated right now, it makes sense to me to go straight for PhD, assuming you are prepared for that. But noticing again that you are not yet a senior, that decision should probably be made a little later, when you know more how much love you still possesses for "the life".

 

[mathwonk]

by the way, in line of fascinating stuff about convergence, you might take a look at "Counterexamples in Analysis" by Gelbaum and Olmstead. Incredible kooky examples in there. Lots of fun.

 

[dkotschessaa]

Yes, I'm planning way ahead right now. Basically information gathering.

Semester is wrapping up. Going to look at some of this stuff we've been talking about over the summer, and hopefully do some tutoring as well.

-DaveK

 

[nickadams]

Do you guys think "working memory" is what determines ones' math ability? Working memory is defined as "a brain system that provides temporary storage and manipulation of the information necessary for such complex cognitive tasks as language comprehension, learning, and reasoning."

http://www.ncbi.nlm.nih.gov/pubmed/1736359

http://www.nytimes.com/2012/04/22/magazine/can-you-make-yourself-smarter.html?pagewanted=allMy working memory is very poor and I was wondering if it would be worthwhile to try to improve it? Or will continuing to do math improve it do you think?

Thanks

 

[chiro]

Do you guys think "working memory" is what determines ones' math ability? Working memory is defined as "a brain system that provides temporary storage and manipulation of the information necessary for such complex cognitive tasks as language comprehension, learning, and reasoning."

http://www.ncbi.nlm.nih.gov/pubmed/1736359

http://www.nytimes.com/2012/04/22/magazine/can-you-make-yourself-smarter.html?pagewanted=all

My working memory is very poor and I was wondering if it would be worthwhile to try to improve it? Or will continuing to do math improve it do you think?

Thanks

I don't know about working memory, but I would say that if you push yourself to as far as you can go personally, then you will probably be very surprised how far you actually get.

The thing about learning and memory per se is that there is no real consensus on both in terms of how they work, why they work and so on.

Sure there are little bits of insight here and there, but the thing is that it's not something that is easy to generalize in a simple way as of yet and if there was (especially for learning), and it was known then teachers and pretty much everyone in general wouldn't be arguing and debating and the process of learning would be very much streamlined.

I know that there are things like the IQ workouts and so on, but really if you want to develop a skill you got to work at it period and for mathematics this meanings thinking about mathematics, reading mathematics, doing mathematics, talking to other people about mathematics and basically expending time and energy in some way on things related to a particular focus of mathematics.

But even then, the thing is also that if you isolate yourself too much on what you 'think' mathematics is vs what mathematics actually is in all its unbounded context, then I personally think you will be missing a large part of the picture.

When you see the entire world through your mathematical lense I gaurantee you will see things that you won't see in greek letter equations in a textbook or formal proofs. It's important to realize this because it's amazing how much is out there and if you spend all your time trying to look for the answers only in one place, then you will probably be missing out on a lot.

Also with regard to comprehension, if you want to improve that then comprehend. One recommendation I have is to answer questions that people ask in the forums: this is a great way to improve comprehension of a subject.

With language, my best suggestion is to read (and read widely) as well as to write. Anything that forces you to organize, plan, and execute your thoughts for different audiences will help you immensely in this regard. Don't just read stuff by the same author or in the same style: read things with many styles and many themes. Listen to a wide range of people who organize and portray their thoughts differently. Force yourself to take the time to purposely have to comprehend something specifically for that person.

As for reasoning, again pay attention to how people reason and not just one group of people. Look at how layman reason, how mathematicians/statisticians reason, how lawyers reason, and how people who have been doing something for many many years reason about things that they have been involved with for a long time.

You can get some good guidelines from mathematics, statistics, logic and philosophy, but remember that if you want some good advice and good reasoning about something, ask someone who has been doing it for a while and is actively engaged in something. The thing is that an expert will be able to see what's really relevant and even if you had good reasoning skills, reasoning on assumptions that are either invalid or completely unknown to yourself is not much use. Also be aware of uncertainty and it's role in reasoning and how you treat reasoning.

 

[battousai]

http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra

If i successfully complete the course above, will I be ready for graduate level algebra?

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