Undergrad mathematics major - what kind of jobs might I end up with?

[colonelcrayon]

I'm a junior in high school looking at colleges. I've recently become very interested in the idea of a mathematics major. My research has led me to the usual suspects, and I've heard that math majors often work in the following:

Actuary
CompSci
Education
Finance
Mathematician
Operations Research
Software Eng.
Statistician

The thing is, I am uncomfortable with the idea of many of these jobs. I have no interest in teaching and statistical analysis bores me (my dad is a statistician, so I've seen that first-hand). I've done a little programming and it can be fun, but I don't think I'd like to do it for 40+ hours a week.

While I like all the math I've done (up through AP Calc AB), I especially enjoyed the following:

solving functions algebraically
proofs in geometry (particularly this)
almost all of calculus

Would a math major be a good way to pursue this sort of work? Or might engineering be a better fit? I've ordered some books introducing more theoretical/advanced math, so I should be able to get a feel for what it's like...

 

[snipez90]

In general, college mathematics is much different from high school mathematics. The emphasis is on rigorous proof and not on computation. By solving functions algebraically, do you mean functional equations? If you liked proofs in geometry, imagine that setting expanded greatly. If you did "two-column" proofs, you will have to get used to more general methods of proof and a less rigid (or sometimes less straightforward) style of writing them. But if you really liked high school geometry, that could be a very good thing; Euclidean geometry has a very well-developed theory, much like various branches of higher mathematics.

As for your interest in calculus, you will likely encounter many more rigorous proofs than you did in Calculus AB. The mathematical branch of analysis is the natural generlization of the various limiting processes you learned in calculus. In a sense, calculus IS analysis, you usually can't tell from a high school calculus course. Calculus is really about estimation (often by means of inequalities), although you are often simply presented with nice formulas in AP Calculus.

May I know which books you ordered?

 

[colonelcrayon]

In general, college mathematics is much different from high school mathematics. The emphasis is on rigorous proof and not on computation. By solving functions algebraically, do you mean functional equations? If you liked proofs in geometry, imagine that setting expanded greatly. If you did "two-column" proofs, you will have to get used to more general methods of proof and a less rigid (or sometimes less straightforward) style of writing them. But if you really liked high school geometry, that could be a very good thing; Euclidean geometry has a very well-developed theory, much like various branches of higher mathematics.

As for your interest in calculus, you will likely encounter many more rigorous proofs than you did in Calculus AB. The mathematical branch of analysis is the natural generlization of the various limiting processes you learned in calculus. In a sense, calculus IS analysis, you usually can't tell from a high school calculus course. Calculus is really about estimation (often by means of inequalities), although you are often simply presented with nice formulas in AP Calculus.

May I know which books you ordered?

I did some basic functional equations, but I was thinking more broadly about working functions out algebraically. It appeals to my analytical, OC side :)

I'll admit that the rigidity of two-column proofs was also appealing for that reason, but I found the satisfaction of finally working out how to prove things about figures in geometry unparalleled. Something is just appealing about geometry, although it helped that I had a great teacher.

I ordered the following:

How to Solve It: A New Aspect of Mathematical Method (G. Polya) - I heard very positive things about this problem solving book, both here and elsewhere
Concepts of Modern Mathematics (Ian Stewart) - I read some of this online and found the style approachable and the content fascinating
Mathematics for the Nonmathematician (Morris Kline) - supposed to provide a look at the history and derivation of many mathematical concepts

I also found the following available in the public domain online:

Mathematics for High School, Elementary Functions (Part 1) - part of the SMSG initiative, should provide some of the background on set theory missing from my school curriculum
Analytic Geometry - another SMSG book
...and any other appealing SMSG items here.

 

[mrb]

There is a chance you would like to be a mathematician. I also really enjoyed my proof-based geometry class in school. After that class it was back to factoring polynomials or whatever mindless nonsense we were doing. It wasn't until pretty late in my college career that I realized I could actually do math (real math like the geometry class, not factoring polynomials) for a living. Now I am (hopefully) on that path.

What you did in that geometry class (try to figure out new facts, and prove that they are true) is basically what mathematicians do.

Unfortunately, if your college math program is set up like mine, you won't actually see any real math until your junior year.

The books you listed look good. In particular I have also heard good things about Polya's book; I have a couple of his other books, but I haven't gotten around to reading them yet.

However, the other two books are just popularizations. These are perfectly good to read and may give you a hint of what real math is but you're probably not going to actually learn any math from them. Perhaps you should look into a serious but basic math textbook. I know three that *might* be appropriate for you:

Calculus by Spivak. It gives a rigorous presentation of Calculus, using material that at most schools is not presented until the "advanced calculus" or "real analysis" level. I really love this book, but it is uncompromisingly difficult. It is very easy to get frustrated with it.

A Course of Pure Mathematics by Hardy. This is another, classic, rigorous presentation of Calculus. Actually the derivative and the integral get less coverage than in other books, under the assumption that the student will get plenty of that elsewhere.

Euclidean and non-Euclidean Geometries by Greenberg. This one, at least in the beginning, is probably much less difficult than the other two. It will build on your knowledge from your geometry class.

These are perhaps too ambitious for you. I don't know. In any case, if you do go this route, don't get bogged down right now. It's OK to just read through the books and do a problem here and there. (Of course, if you do decide that this is the path for you, you will have to get down to work, and then the best procedure is usually to do almost all the problems.)

 

[colonelcrayon]

^ Interesting. For the moment, I'm working through Gilbert Strang's calculus textbook (free from MIT) and it's already much more interesting than the standard AP Calc curriculum. It shows a bit more of the relationship between calculus and other math.