Preparation for heading into college as a math major

[WillJ]

I'm about to graduate from high school, and in college I plan to be a math major (with another major in economics).

I was thinking that this summer I should do something to prepare a bit for that.

My senior year I have taken IB SL Calculus (roughly equivalent to AP Calculus AB), and I will repeat Calculus I my first year in college. I could take Calculus I at my local college in the summer and immediately go to Calc II in college, but I don't think that makes much sense. I think it'd make a lot more sense to spend time building solid, general math skills, rather than specifically learning the subject of calculus. I'm looking for a way to accomplish the following two things:

1. Solidify my knowledge of precalculus topics (algebra, geometry, trigonometry) --- refreshing the small bits that I may have forgotten (especially in geometry), as well as perhaps going a bit beyond what I've already done.

2. Start training myself to see math through a mathematician's eyes, which means the following things: Exercise my mathematical creativity. Learn the basics of writing proofs. Get used to approaching problems that I haven't been explicitly taught how to solve before (and, of course, have fun doing so).

Do you guys agree my plan makes sense? And if so, what do you think is the best way of doing this? I looked at my local college to see if they have a course that is designed specifically to do this, but they don't, so it looks like this will have to be self-taught. I'm hoping, though, that it won't have to be too self-taught --- perhaps there is a book out there made just for someone in my situation, or something. (I wouldn't want to just browse the internet for random problems to solve; I'd like this process to be somewhat cohesive.)

 

[JeffN]

when i was in your position last year, I was recommended this book:

https://www.amazon.com/dp/0521675995/?tag=pfamazon01-20

i didn't end up finishing it, but the parts that i did get through helped me get familiar with university level math. The book doesn't really deal with the precalculus stuff that you mention, but should help you get more comfortable with proofs.

 

[trinitron]

https://www.amazon.com/dp/0195105192/?tag=pfamazon01-20

 

[Brad Barker]

sounds like you have a good idea of what math takes.

the more time you spend being rigorous with your earlier class, the easier you'll find reading the textbooks for the proof-y classes.


can't comment on any of the aforementioned books. at UF, we have two classes specifically designed for forming proof writing skills. i hope that it's the same at other schools. if not, check those books out.

 

[gravenewworld]

Don't waste any money taking Calc I or II or any other math class during the summer because you are just going to pay for it again when you take it at your university, and the credits may not even be transferable!

You should be spending your summer after you graduate high school having fun, not trying to do math proofs that you are probably not going to do in college. Trust me you probably don't have the slightest idea of what a limit is yet, you won't learn that until you are probably a sophomore or junior when you get to do real analysis.

The most important thing to be good at coming out of high school is algebra. I really didn't even use trig that much at all when I doing my math degree since all the trig functions can easily be expressed through complex numbers. If you are good at algebra you shouldn't have any trouble at all starting out as a math major. Just keep those skills sharp by doing 10 or so algebra problems a week.

Since you want to study economics too, I would suggest that you should be more concerned about tailoring your schedule towards supporting your economics degree. YOU SHOULD DEFINITELY TAKE STATISTICS if you want an economics degree. Courses like econometrics use heavy amounts of statistics. If you are interested in econ you should seriously take as many stat, linear algebra, analysis, and differential equations classes as possible. Learning stochastic methods also helps out too.

 

[JeffN]

what gravenewworld said about having fun is completely true, especially if you're moving away for school.

 

[Mandanesss]

You're graduating high school, you've worked hard for four years and about to embark on another four years of intense studying, take a break. I wouldn't be doing proofs this summer if I were you. Highschool calculus is a lot different than university level calculus. But I agree that you should definitely review basic algebra. I was never good at algebra, so I just started from scratch with Intermediate algebra in college and it has made all the difference. If you have a good foundation in algebra, you can definitely finish problems faster and with ease. Also, I would brush up on trig functions and know the graphs of them like the back of your hand. Other than that, enjoy your summer before college.

 

[J77]

I'd third taking a break and add...

Learning higher level stuff from a textbook, over the summer, then starting uni and finding it taught completely differently, is a recipe for confusion/disaster/low morale.

Read some pop-science books but don't go in for hardcore study.

 

[MathematicalPhysicist]

he's having fun cracking up his head over maths problems, what's the harm in that?!

 

[J77]

he's having fun cracking up his head over maths problems, what's the harm in that?!

One point would be that the older you get, the less and less time you get for having extended holidays.

Between high school and uni is a lovely extended break on which you can travel or work, out there in the big world.

 

[WillJ]

Okay guys, you're probably right; it'd probably be best to just take a nice, long break.

About algebra: As far as I know, I'm not really rusty in algebra at all (especially since I'm taking both calculus and physics and using plenty of it). Of course, I could be wrong about that. Can you give a few examples of algebraic problems/topics that I should know how to do but might have forgotten?

Edit: I guess one thing in algebra I'm weak in is matrices (since my teacher never taught us the logic behind them in depth, but just had us mindlessly do problems related to them). But I don't have to worry about that, since I'll learn all about that in linear algebra class, right?

 

[mathwonk]

i agree with trinitron's recommendation of courant and robbins, what is mathematics?

some of us enjoy math, done well and intelligently. that in itself is a break from rote math courses in high school, like ap calc.

i especially recommnend something like the promys program at BU, for bright high schoolers, which teaches them math from a mathematicians viewpoint, through number theory.

that should be fun and exactly what you asked for in content.

 

[gravenewworld]

Have you ever thought about just minoring in economics? Econ grad departments KILL for students with extremely strong math backgrounds. Having a math degree puts you at a SIGNIFICANT advantage over regular econ students who apply to grad school. The reason I say just think about getting a minor in economics is because a minor allows you to take more math courses! Econ grad departments care much much more about how much math you have taken rather than how much econ you have taken because econ grad school can teach you economics, not math. Honestly, you don't even really learn real economics as an undergrad, just some basic principles. To even see what grad school econ is like, you would need to take a course on advanced macro/micro economics. Absolutely 0 economists use stuff like the IS/LM model that is taught to undergrads to characterize the economy. You don't need an econ degree at all to pursue economics in grad school.I majored in math and minored in econ. I was a hair away from applying to grad school for economics, I had all my recommendation letters ready, took the GRE, and filled out all the applications. In the end though, I just chose not to send it in.

The more math you take, the better it is for econ grad school. Econ departments love to see a strong back ground in analysis and stat. Completing a course on functional analysis with a good grade almost guarantees you a spot in econ grad school at a good university.

 

[mathwonk]

and economists make WAY more money than math profs.

 

[Mandanesss]

gravenewworld, what's a functional analysis class? Is that also called real analysis?

 

[gravenewworld]

gravenewworld, what's a functional analysis class? Is that also called real analysis?

No, something like the study of hilbert spaces would be functional analysis. You should definitely study advanced calc before taking functional analysis.

 

[gravenewworld]

Okay guys, you're probably right; it'd probably be best to just take a nice, long break.

About algebra: As far as I know, I'm not really rusty in algebra at all (especially since I'm taking both calculus and physics and using plenty of it). Of course, I could be wrong about that. Can you give a few examples of algebraic problems/topics that I should know how to do but might have forgotten?

Edit: I guess one thing in algebra I'm weak in is matrices (since my teacher never taught us the logic behind them in depth, but just had us mindlessly do problems related to them). But I don't have to worry about that, since I'll learn all about that in linear algebra class, right?

knowing all the tricks in high school algebra will save you time. know stuff like

-knowing your p's and q's to find the roots of a polynomial

-all the log tricks

-horner's method

-all the standard graphs and how they are "moved" around by playing around with the function

-completing the square

-coming up with equations for a line. y1-y2=m(x1-x2) for some reason is a favorite of professors, not just math professors, but science profs. also.

etc.

I swear, I never used any trig more than sin^2+cos^2=1 or a double angle formula once in a while my entire career as a math major and you know what, I never took trig before in my life, I skipped it in high school. I still did fine. College math is NOT going to test your trig abilities very much, that is what high school was for. If there was an identity I needed I just derived it from complex numbers if it wasn't too time consuming or just looked it up in a table. don't worry about matrices. you will learn them later.

 

[mathwonk]

functional analysis is that part of real and complx analysis involving infinite dimensional spaces.

 

[WillJ]

i agree with trinitron's recommendation of courant and robbins, what is mathematics?

some of us enjoy math, done well and intelligently. that in itself is a break from rote math courses in high school, like ap calc.

i especially recommnend something like the promys program at BU, for bright high schoolers, which teaches them math from a mathematicians viewpoint, through number theory.

that should be fun and exactly what you asked for in content.

I checked out PROMYS, and it looks like I wouldn't be able to do it since it overlaps with some dates on which I have to do stuff for Vanderbilt (the college that I'll be attending).

Have you ever thought about just minoring in economics? Econ grad departments KILL for students with extremely strong math backgrounds. Having a math degree puts you at a SIGNIFICANT advantage over regular econ students who apply to grad school. The reason I say just think about getting a minor in economics is because a minor allows you to take more math courses! Econ grad departments care much much more about how much math you have taken rather than how much econ you have taken because econ grad school can teach you economics, not math. Honestly, you don't even really learn real economics as an undergrad, just some basic principles. To even see what grad school econ is like, you would need to take a course on advanced macro/micro economics. Absolutely 0 economists use stuff like the IS/LM model that is taught to undergrads to characterize the economy. You don't need an econ degree at all to pursue economics in grad school.


I majored in math and minored in econ. I was a hair away from applying to grad school for economics, I had all my recommendation letters ready, took the GRE, and filled out all the applications. In the end though, I just chose not to send it in.

The more math you take, the better it is for econ grad school. Econ departments love to see a strong back ground in analysis and stat. Completing a course on functional analysis with a good grade almost guarantees you a spot in econ grad school at a good university.

Yeah, I've thought about doing exactly that ... But at Vanderbilt (the school I'll be attending), I can major in economics with honors by taking the following:

-Intro Micro
-Intro Macro
-Intermediate Micro
-Intermediate Macro
-Econometrics
-a specialty course, like Development Economics (the area I'm most interested in)
-Independent Study
-Honors Thesis

The last two can constitute almost half of my work, and I think think doing independent econ research like that would be both very interesting and very useful for grad school (and I could make my research as "real" / "mathematical" / "grad-school-like" as I wish). And of course, everything above the last two things pretty much constitutes an econ minor, and would be helpful in getting acquainted with economics (to make sure I want to do it in the first place!). If you have any thoughts on that, I'm all ears.

While I'm at it, I imagine I'll be able to take the following math courses (more or less):

-1st year accelerated calculus
-2nd year accelerated calculus
-linear algebra
-differential equations
-intro to mathematical statistics
-intro to applied statistics
-probability
-mathematical statistics
-intensive problem solving and exposition
-intro to mathematical logic
-introduction to analysis
-topology of surfaces
-set theory

(Originally I wanted to take topology and set theory simply because they sound extremely interesting to me, but then I learned they're actually used extensively in economics nowadays, crazily enough.) Of course, these lists are purely tentative, considering I haven't even started college yet.

knowing all the tricks in high school algebra will save you time. know stuff like

---list and stuff---

Okay, I'll check out a couple of those things in-depth (the rest I already feel comfortable with). Also, I'll be taking Vanderbilt's math departmental exam this summer, so I guess I can see what I'm weak in through that.

 

[gravenewworld]

Yeah, I've thought about doing exactly that ... But at Vanderbilt (the school I'll be attending), I can major in economics with honors by taking the following:

-Intro Micro
-Intro Macro
-Intermediate Micro
-Intermediate Macro
-Econometrics
-a specialty course, like Development Economics (the area I'm most interested in)
-Independent Study
-Honors Thesis

The last two can constitute almost half of my work, and I think think doing independent econ research like that would be both very interesting and very useful for grad school (and I could make my research as "real" / "mathematical" / "grad-school-like" as I wish). And of course, everything above the last two things pretty much constitutes an econ minor, and would be helpful in getting acquainted with economics (to make sure I want to do it in the first place!). If you have any thoughts on that, I'm all ears.

A typical minor in econ would be Intro to micro/macro, Intermediate Micro/Macro, and then two upper level courses. That alone would be plenty to get into grad school for econ. For upper levels I would definitely recommend taking Econometrics, advanced macroeconomics, and possibly game theory. With a minor in econ, you still have room to take more econ classes since you will probably have spots open for electives through your math degree. That is exactly what I did. I took all the econ minor requirements and ended up taking about 2 more econ classes than I really needed because I had room for electives. Research is always good. It is all up to you though. Just be aware that you don't need a degree at all in economics to pursue it futher if you wanted to in grad school. If you don't want to go to grad school in economics then you can completely disregard what I posted and go for your econ degree.

While I'm at it, I imagine I'll be able to take the following math courses (more or less):

-1st year accelerated calculus
-2nd year accelerated calculus
-linear algebra
-differential equations
-intro to mathematical statistics
-intro to applied statistics
-probability
-mathematical statistics
-intensive problem solving and exposition
-intro to mathematical logic
-introduction to analysis
-topology of surfaces
-set theory

(Originally I wanted to take topology and set theory simply because they sound extremely interesting to me, but then I learned they're actually used extensively in economics nowadays, crazily enough.) Of course, these lists are purely tentative, considering I haven't even started college yet.

The list looks good for starters. Ever read an economics journal before? Economists use pretty heavy duty math all the time, I wouldn't doubt for a second that they would need to use some ideas from set theory and topology. See if you can squeeze in some more analysis, like complex analysis.

 

[Werg22]

The transition from algebra to analysis takes way more time than a summer, let alone three weeks. If you though yourself calculus II in three weeks then you have missed out on allot. Could you in simple terms define what is a limit? What is continuity? Prove the uniqueness of the limit of a Riemann sum off the principle of uniform continuity? There is much more to learn about analysis than how to "use it". The problem is that, without realizing it, especially if you want to become a math major, you are giving yourself poor bases.

 

[mathwonk]

when you get to vanderbilt, please go and say hello to my student pat thompson.

he seems to have moved across the street to peabody school of education. please say hello if you get a chance. it will be a good introduction for you to the community.

my name is roy smith, from ellensburg.

vanderbilt seems now to have become a very strong place in mathematics. they apparently have a lot of money and are buying the best talent they can find.

nashville is also a very pleasant place naturally.

 

[WillJ]

See if you can squeeze in some more analysis, like complex analysis.

Okay.

The transition from algebra to analysis takes way more time than a summer, let alone three weeks. If you though yourself calculus II in three weeks then you have missed out on allot. Could you in simple terms define what is a limit? What is continuity? Prove the uniqueness of the limit of a Riemann sum off the principle of uniform continuity? There is much more to learn about analysis than how to "use it". The problem is that, without realizing it, especially if you want to become a math major, you are giving yourself poor bases.

Who exactly are you addressing here? Me? I'm not sure why you think I plan to transition myself "from algebra to analysis" in "three weeks" or teach myself calculus II.

when you get to vanderbilt, please go and say hello to my student pat thompson.

he seems to have moved across the street to peabody school of education. please say hello if you get a chance. it will be a good introduction for you to the community.

my name is roy smith, from ellensburg.

vanderbilt seems now to have become a very strong place in mathematics. they apparently have a lot of money and are buying the best talent they can find.

nashville is also a very pleasant place naturally.

Okay, I'll try to do so. Just curious, since I'm confused, what do you mean by him being your student? Do you mean ex-student? Is he somehow doing something math-related in the college of education, or did he "convert"?

 

[mathwonk]

functional analysis means infinite dimensional real analysis.