It's really a mild, soft question.

So far, I am an undergrad student, contemplating on several majors.

What will be the major difference if I become a math major and go to a graduate school to study math?

Will a graduate math major generally focus on reading/studying recent research papers? Or will the graduate math major be taught by professors on the things not covered in undergrad programs?

The reason why I am asking this question is that undergrad math programs seem to me at this point so well-covered that after studying the programs, a person will be able to do anything he wants to do, and if one wants to do research, he will be able to catch up with recent progress by reading research papers.

If this is true, why is the graduate school even needed?

By the way, I am a first-year undergraduate student :)

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So how exactly do you know what one needs to know to read research papers in all areas? –  Michael Greinecker May 22 '12 at 11:57
When you start your grad level studies you take some courses, learn more on some topics, then you read a lot on your own and study more on selected topics, then you can read papers in those topics, and eventually you should be able to come up with new ideas in the field. –  Asaf Karagila May 22 '12 at 11:59
Your impression is not very accurate. As a freshman, perhaps when you see the exotic topics that lie ahead of you think that you will learn a vast amount in your next few years, you think surely that must consist of a large portion of mathematics. Alas, it is but a drop in the ocean, and unless your Gaussian abilities allow you to sail the seven seas, in grad school you will probably pick a little island and live there the rest of your life like most people do. –  Ragib Zaman May 22 '12 at 12:32
Thanks everyone. –  user2346 May 22 '12 at 13:00
In hindsight, I wish I had studied fewer subjects deeply, and that I had kept careful notes of my solutions. It doesn't make sense to invest time in a proof/learning something unless you'll be able to use it later. Most of the solutions I wrote down ended up in the waste bin, because I figured "I could just write it down again." True, but at the cost of probably hundreds of additional hours of work. Also, I spent way too much time trying to understand proofs than actually using results to solve problems; I assumed that every proof I came across was well-written and worth remembering (mistake). –  vgty6h7uij May 23 '12 at 12:53

As a graduate student, the most useful skill I learnt as an undergraduate was not the mathematics itself, but how to learn mathematics. The edge of the subject is so wide that it's mostly not practical to get to a lot of current research problems as an undergraduate, even in a particular subfield like geometry or algebra for example.

That's not to say that the mathematics isn't important (and in fact I'm probably underplaying its importance because the parts of it I use all the time have become second nature), but knowing how to learn things efficiently is incredibly useful.

The graduate school experience probably varies quite a lot from university to university (and between countries as well), but my experience is similar to that described by Asaf in the comments - you still do some more formal courses at the start, but more independently than as an undergraduate, and at the same time your supervisor will suggest things you should read and problems you should think about - and these should lead to you discovering more things to read and problems to think about under your own volition.

I should probably also recount what I've always heard said by lecturers as the big difference between begin an undergrad and a grad student - as a grad student, you have to contribute original research. The upshot of this is that while the problems you see as an undergraduate may be difficult, they at least have answers, but this need not remain true when you are a grad student, and learning how to make judgements about which questions are worth persuing is an important aspect of postgraduate study - as mentioned by Eugene, Terence Tao has lots of good advice along these lines.

This also leads to a sound-bite answer to your question "why is graduate school even needed?" - because the process of learning how to do research is distinct from the process of learning how to do mathematics.

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Here is what I tell my grad students:

The difference between undergrad mathematics and graduate mathematics is the difference between art history, or art appreciation, and learning to be an artist.

As an undergraduate you see a lot of mathematics, but you don't create new mathematics. The goal of graduate school (and here I am speaking from experience with top fifty U.S. graduate schools, so what I am saying probably applies best in that context) is to learn how to create new mathematics, and then to create that new mathematics.

One specific consequence of this (in my view) is the following: often in undergraduate mathematics classes, proofs and rigor are presented almost as moral imperatives --- as if it is a moral failing to know a statement without knowing why it is true; consequently, people often put a lot of effort into learning arguments just for the sake of having learnt them. (This is exaggerated, perhaps, but I think it reflects something real.) On the other hand, in research, one learns arguments for different reasons: to learn technique, to pick out important ideas --- there is a professional aspect to the way one looks at pieces of mathematics which is not usually present in undergraduate mathematics. One gives proofs in order to be sure that one hasn't blundered; one's interaction with the mathematics and the arguments is much more visceral than in undergraduate courses.

(I am not speaking from any experience now, but I think of the difference between learning how to interact with a block of marble, and bring a new form out of it, however rough it might be, in comparison to looking and learning about a lot of existing beautiful statues, masterpieces that they are.)

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Thank you for posting this, by the way. –  Akhil Mathew May 26 '12 at 0:23
To claim that you aren't really doing math until you're doing research belittles legitimate math that undergraduates do. –  Andrew Kelley Mar 30 '14 at 1:33
Dear Andrew, This post is discussing the creation of new mathematics. This is the focus of graduate training in mathematics, and is not the focus of undergraduate training in mathematics, which is instead focused on learning existing mathematics. (At least this is my experience of the two. Perhaps your is different.) Regards, –  Matt E Mar 30 '14 at 4:10

Terence Tao has great advice for mathematicians at every stage of their careers.

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To get an idea of how mathematics graduate school in the United States works I would suggest the book A Mathematicians Survival Guide by Steven Krantz. The books is filled with information about the ins and outs of mathematics graduate school, including advice concerning many of the common pitfalls experienced by graduate students. It also contains advice for recent PhDs about their options after graduate. Overall, a very useful source for those unsure of whether graduate school is for them. The preview on Amazon should give you a good idea about the contents of the book.

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From one who failed to realize all the value of his own education... and to provide a wider perspective to @MattPressland's excellent answer.

Although written for high school students, I believe that What You'll Wish You'd Known by Paul Graham resonates deeply with your question. Specifically in wondering "why is the graduate school even needed", I think the following quote sums the essay and answers your question:

Suppose you're a college freshman deciding whether to major in math or economics. Well, math will give you more options: you can go into almost any field from math. If you major in math it will be easy to get into grad school in economics, but if you major in economics it will be hard to get into grad school in math.

This applies to graduate school in that having a Masters in Mathematics will provide you with more options than a bachelors alone. Although theory is often not as applicable in practice, I'd certainly hire the PHD who could show the output of their labor over the BA who could demonstrate the same.

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User2346:

Here's the opinion of someone who majored in math (at a top school in the United States), but was not a genius at it.

This is a condensed, generalized, down-and-dirty opinion, but I stand by it.

Your assumption about being well-covered is more accurate that not. If your intent is to form a mathematical basis of strength that you can apply to applications of math (e.g. finance stat econ) then undergrad is as far as you need to go with it. By the time you take Real Analysis & Complex analysis your mind will have developed in an amazing way. As Matt Pressland said, you learn how to learn Mathematics. Matt also mention the difference between that style of meta-cognition and being able to research. It's impossible to say today that you could or couldn't come up with a unique research idea that would be considered a contribution and that you're passionate about to pursue. However, unless the word research makes your heart throb, you don't need Math graduate school.

I've rewritten the rest of my answer to be a bit more compact.

1. Do not major in math, minor in it. You will gain a superior analytic mind. However, your employment potential depends on not being a Math major. The Corporate world (Actuarial aside) doesn't know wtf Math is. They will ask if you wanted to Teach High School.

2. Accounting is boring, and Economics is fluffy. Choose Statistics and augment with Finance. Also, Computer Science requires more hours spent at 1-sitting than Mathematics. Engineering trumps Computer Science as you can learn to be a programmer in your spare time.

3. First know what you want to do as a career. Do not decide this on your own. Go to your career center for expert help, and shadow professionals and intern. Funny how this is #3.

Conclusion Math can be a minor. Let your possible career path determine your major. At the least you can be earning a salary when you're 22-23 while deciding what new career you want. At the best, you'll save yourself 3-5 years of searching, finding-professional-self, and rebuilding qualifications.

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I decided to add some external links that would shed light on why some of my views are as they are. These come fro the Bureau of Labor Statistics... Oh look Statistics is even in the name =)

A small summary is included with the link, but please click and compare. These are just samples, you can find your own BLS data to refute if you want.

Occupations from mostly-pure discipline

1. Economist. $43/hr. 6% projected growth. 15.4K jobs. Bachelors Entry-level http://www.bls.gov/ooh/Life-Physical-and-Social-Science/Economists.htm 2. Mathematician.$48/hr. 16% projected growth. 3.1K jobs. Masters Entry-level
http://www.bls.gov/ooh/math/mathematicians.htm

3. Statistician. $35/hr. 14% projected growth. 21.5K jobs. Masters Entry-level http://www.bls.gov/ooh/Math/Statisticians.htm  Occupations heavy in STAT with Finance Augment 4. Financial Analyst.$35/hr. 26% projected growth. 236.0K jobs. Bachelors Entry-level

5. Actuary. \$42/hr. 27% projected growth. 21.7k jobs. Bachelors Entry-level
http://www.bls.gov/ooh/math/actuaries.htm


Conclusion for the sample

Stat has way more job positions filled/needed much faster growth than Econ, same entry-level as Math.

Finance augmented, Statistically heavy jobs are quite abundant, pay well, and grow fast. This is what I recommended. I recommended majoring in Stat with a Finance augment. Both examples of this require only a bachelors and are in a job space projected to grow much more than the Mathematician or the Economist. The pure Statistician indeed makes less than those two. So to repeat from previously: Math Masters not needed if you like other things. Don't major in Math in undergrad.

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I have to say that my classmates who were taking math as a major and another minor, or worse: taking math as a minor, we're not exposed to enough mathematics and it is incredibly difficult for them to pursue a masters degree in mathematics. If one intends to go to grad school in mathematics, one should learn mathematics -not just see a review of it. –  Asaf Karagila May 23 '12 at 6:16
Agreed. The masters in Math has to be the goal though. –  VISQL May 23 '12 at 16:46
I really disagree with this answer, but I'm not going to down vote it. Saying a bunch of blanket statements such as "accounting is boring, economics is fluffy, engineering trumps computer science,..." is not well taken. –  Samuel Reid May 24 '12 at 0:39
@Samuel. I understand but there's no time for a lecture. Go to BLS.gov and do a search for Mathematician, Statistician, Economist, Financial Analyst and then look back to what I say. –  VISQL May 24 '12 at 16:40
Please take time to refute my comment with more evidence than I present or to look into yourself before down voting, which is very negative mostly impulsive reaction to the opinion of another based on your own opinion. –  VISQL May 24 '12 at 17:08