High School Math vs College Math?

[wlcgeek]

Hey everyone,

Just wondering in what ways do high school math differ from college level math?

Thanks.

(Also, are there any recommended books or other materials to help a person better make that transition?)

 

[micromass]

It's not clear to me what you mean with "college-level math". This is because math in college can be done in a large number of ways.

For example, if you study engineering, then you will find that mathematics is not significantly different in college. It's more difficult and you'll see more mathematics in the same time period, but it largely remains very computational.

In pure mathematics, there is a very significant change. Computations become much less important. The focus lies on proofs and abstractions. Instead of focusing yourself on the real numbers, like in HS, you will meet several generalizations and abstractions. If you liked math in HS, then there is no real guarantee that you will like pure math.

I guess most college mathematics falls between these two extremes. There are very computational courses, and very abstract courses.

So, can you tell me what you are planning to study in college? I might be able to give you more information about which books are good to go through.

 

[Fredrik]

I think the main difference (assuming that we're talking about pure mathematics, and not some kind of applied mathematics) is that mathematics at the university level is based on set theory. (The first few courses you take may try to hide that from you, but they can't do that forever). Fortunately for me, the first course we took covered the basics of logic (logical symbols and truth tables) and naive set theory (how to use the notation, and develop an intuitive understanding of sets). Then our teachers at the second and third math courses refreshed our memories about some of these things at the start of those courses.

It seems to me that far from everyone gets that kind of introduction, and that they struggle as a result of it. So one way to prepare would be to begin to study these things in advance. However, it seems to be quite hard for people at the high school level to do that, because of their lack of experience with functions and proofs.

 

[micromass]

I think the main difference (assuming that we're talking about pure mathematics, and not some kind of applied mathematics) is that mathematics at the university level is based on set theory. (The first few courses you take may try to hide that from you, but they can't do that forever). Fortunately for me, the first course we took covered the basics of logic (logical symbols and truth tables) and naive set theory (how to use the notation, and develop an intuitive understanding of sets). Then our teachers at the second and third math courses refreshed our memories about some of these things at the start of those courses.

It seems to me that far from everyone gets that kind of introduction, and that they struggle as a result of it. So one way to prepare would be to begin to study these things in advance. However, it seems to be quite hard for people at the high school level to do that, because of their lack of experience with functions and proofs.

Very true. I never really had significant troubles in my undergrad, and I think that is due to our education system here. We actually saw set theory in elementary school! Well, not formal proofs of course, but they made us familiar with things like intersections and unions on a very basic level. So by the time I got to university, the language wasn't alien to me.

Here in my country, a freshman math student starts off by taking real analysis and abstract algebra. I was quite shocked the first time I found out that these are considered very difficult courses. Since I don't think that American students are less capable than European students, I think the only conclusion one can make is that the high school education (and before) is severely lacking in most US schools. For example, set theory and logic is never really covered there.

Anyway, what I want to say that it might be very benificial to learn basic set theory and basic logic. This will help you a lot if you study pure mathematics. That said, it is a rather boring subject, so don't be put off if you don't like it very much.

 

[hsetennis]

Hey everyone,

Just wondering in what ways do high school math differ from college level math?

Thanks.

(Also, are there any recommended books or other materials to help a person better make that transition?)

Assuming you refer to pure mathematics, Ethan D. Bloch's Proofs and Fundamentals, A first course in abstract mathematics, is a good place to start.

 

[wlcgeek]

It's not clear to me what you mean with "college-level math". This is because math in college can be done in a large number of ways.

For example, if you study engineering, then you will find that mathematics is not significantly different in college. It's more difficult and you'll see more mathematics in the same time period, but it largely remains very computational.

In pure mathematics, there is a very significant change. Computations become much less important. The focus lies on proofs and abstractions. Instead of focusing yourself on the real numbers, like in HS, you will meet several generalizations and abstractions. If you liked math in HS, then there is no real guarantee that you will like pure math.

I guess most college mathematics falls between these two extremes. There are very computational courses, and very abstract courses.

So, can you tell me what you are planning to study in college? I might be able to give you more information about which books are good to go through.

I want to at least minor in math and possibly major in it with something else (not sure what yet). :tongue: I enjoy physics and English currently as well.

The main thing is that I have heard math is just different in college. Supposedly professors or the work itself is not as much memorization as in high school. In hs, math is not too difficult, due to just pattern recognition, memorization, and a little bit of problem-solving.

People have told me that college math is much more heavy on problem-solving and you don't get exam questions where you just plug-and-chug or just run through a series of memorized steps.

Why do you guys make a distinction between "pure" math and other types of math? Why is it more pure than high school math? lol

 

[wlcgeek]

Assuming you refer to pure mathematics, Ethan D. Bloch's Proofs and Fundamentals, A first course in abstract mathematics, is a good place to start.

Thanks. I will check if my library has it!

Does it teach from the perspective of a math major already or does it speak to a more general audience? I think I want a beginner book that an average person can read, but still retains the necessary complexity/quality of ideas.

 

[Naty1]

I want to at least minor in math and possibly major in it with something else (not sure what yet).

engineering math is applied mathematics...how to use math; as noted, pure math is more abstract and theoretical...

I studied electrical engineering [in New York City] many years ago and took an extra Calculus math class just because I was interested...In the last semester of my senior year of EE I found out that had I taken only two or three additional math classes I could have also earned as BS in math...by then it was too late! So a 'dual degree' may also be possible...

A college math experience:
My EE calculus professor for three semesters wrote with his right hand and erased with his left moments later! ...I don't believe to this day he ever stopped even for a moment during class. I am not sure he even took a breath. I did every problem in the text...[Thomas..Calculus and analytic Geometry...that text is still around today]... you had to be VERY alert and a fast note taker. My first semester I got a 'c'...I was really disappointed...until I later found about 10% or 15% of the class had already semester of Calculus in high school... so for many it was review! After that I got A's.

Take all the math [and/or science] you can in HS and college... if you like it ...it will always distinguish you from the history, philosophy, language, etc majors when job hunting...

 

[hsetennis]

Thanks. I will check if my library has it!

Does it teach from the perspective of a math major already or does it speak to a more general audience? I think I want a beginner book that an average person can read, but still retains the necessary complexity/quality of ideas.

I don't know of any public libraries holding college mathematics textbooks (I may be wrong). Perhaps some university libraries may have it.

It and most other "transition/intro to upper math" textbooks are not meant for the general public. These kinds of books have very simple and straightforward exposition, but they assume that you have taken at least a few semesters of calculus so that you are ready for more the theoretic classes like algebra and analysis.

 

[wlcgeek]

Hi guys,

Thanks again and I have a follow-up question.

In high school there is a "ranking/ordering" of math (Pre-Alg, Alg I, Alg II/Geo, Trig, Calc...). I'm wondering if there is a "hierachy" of math at higher levels? I'm sure some math builds on others and some may be independent, but just curious if anyone can list (or link to) a ordering of all the types of math a person might encounter?

Hope that makes sense!

 

[hsetennis]

Hi guys,

Thanks again and I have a follow-up question.

In high school there is a "ranking/ordering" of math (Pre-Alg, Alg I, Alg II/Geo, Trig, Calc...). I'm wondering if there is a "hierachy" of math at higher levels? I'm sure some math builds on others and some may be independent, but just curious if anyone can list (or link to) a ordering of all the types of math a person might encounter?

Hope that makes sense!

The system is fairly similar at most universities. You can find suggested course paths on the webpage of math departments at any large institution.

 

[DuncanM]

Hey everyone,

Just wondering in what ways do high school math differ from college level math?
. . .

In addition to being taught at a much higher level, college/university math courses are usually totally different subjects.

High school laid the ground work for the material you will learn in college. In high school, I covered basic algebra, some trigonometric identities, logarithms, maybe a brief introduction to Calculus.

In university, everybody takes Calculus in first year. And even if you take some in high school, the problems are a lot harder, and you are often asked to show a proof or derivation of some concept. You are not just given an equation and expected to plug in numbers.

Plus, the courses are totally different and specialized for different mathematical fields:
Calculus, Linear Algebra, Complex Numbers, Topology, Differential Equations, Discrete Math, and many more.

 

[wlcgeek]

I think the main difference (assuming that we're talking about pure mathematics, and not some kind of applied mathematics) is that mathematics at the university level is based on set theory. (The first few courses you take may try to hide that from you, but they can't do that forever). Fortunately for me, the first course we took covered the basics of logic (logical symbols and truth tables) and naive set theory (how to use the notation, and develop an intuitive understanding of sets). Then our teachers at the second and third math courses refreshed our memories about some of these things at the start of those courses.

It seems to me that far from everyone gets that kind of introduction, and that they struggle as a result of it. So one way to prepare would be to begin to study these things in advance. However, it seems to be quite hard for people at the high school level to do that, because of their lack of experience with functions and proofs.

I've learned set builder notation in high school, but I'm not sure about set theory.

Can you guys recommend any books on set theory or logic that I can read on my own to have this type of introduction?

Thanks again!

 

[Fredrik]

I've learned set builder notation in high school, but I'm not sure about set theory.

Can you guys recommend any books on set theory or logic that I can read on my own to have this type of introduction?

Thanks again!

If you want a quick summary of the basics, then you can try this pdf. If you want an introduction that's a bit longer and doesn't go into all the gory details of axiomatic set theory, then I think an introduction to proofs books is right for you, for example this one. I see that hsetennis recommended something like that above, so you should check out his recommendation too.

If you want a book on axiomatic set theory, then Hrbacek & Jech is probably the best one. If you are especially interested in the set theoretic definitions of the number systems, then Goldrei is an alternative.

Not sure how much you will want to know from logic. You can get pretty far just by understanding the basic notation and truth tables. This sort of stuff is covered in those introduction to proofs books.

 

[hsetennis]

I've learned set builder notation in high school, but I'm not sure about set theory.

Can you guys recommend any books on set theory or logic that I can read on my own to have this type of introduction?

Thanks again!

You probably don't need a whole book to pick up basic set theory and logic, unless you want to go really deep into it. Many upper-level undergrad books will have a sufficient introduction.

 

[phosgene]

How to Prove It by Daniel J. Velleman is a great intro to set theory and mathematical logic. It only assumes basic high school maths, but the contents are *very* helpful for when you encounter proofs later on.

 

[lntz]

Very true. I never really had significant troubles in my undergrad, and I think that is due to our education system here. We actually saw set theory in elementary school! Well, not formal proofs of course, but they made us familiar with things like intersections and unions on a very basic level. So by the time I got to university, the language wasn't alien to me.

Here in my country, a freshman math student starts off by taking real analysis and abstract algebra. I was quite shocked the first time I found out that these are considered very difficult courses. Since I don't think that American students are less capable than European students, I think the only conclusion one can make is that the high school education (and before) is severely lacking in most US schools. For example, set theory and logic is never really covered there.

Anyway, what I want to say that it might be very benificial to learn basic set theory and basic logic. This will help you a lot if you study pure mathematics. That said, it is a rather boring subject, so don't be put off if you don't like it very much.

I'm not especially familiar with set theory, but from what you have said, perhaps it's just never been called that when we use it to discuss other things.

Quick question, is Set theory what Frege worked on and where Russel's librarian paradox comes from?

I only know the "Tale" rather than the math behind that, but it's a great paradox :)

 

[billyn00b_2001]

micromas u are 16 and u already went to college wow

 

[mathwonk]

It varies from college to college, but one difference that can certainly occur is that the student in college needs to take far more responsibility for his own learning. In some high schools all you need to do is attend class and do the homework problems which are mainly easy computations. And the teacher may go rather slowly and even repeat the material over and over until most students grasp it.

In college it is not unheard of for the professor to cover each topic only once, or even at most once. Whether you learn it or not with that one presentation is up to you. I.e. you are expected to take the notes and the book and any other sources you need, and go home and to the library and work with your friends and just make sure you learn it. I.e. the professor just presents it once, you have to do the repetitions you need to learn it yourself. This one difference is the downfall of most students I had in college, they just did not take responsibility for their own learning. In particular they often declined to come to office hours unless there was a test or assignment due the next day, or even the same day. Try not to be like that.

In reference to what micromass and Fredrik mentioned, the abstract approach to math in college, I had a course from this book in high school that helped me a lot.

http://www.abebooks.com/servlet/Sea...er,+oakley&sts=t&tn=principles+of+mathematics

It starts off with basic logic and propositional calculus, , then discusses numbers systems including complex numbers correctly, and goes on to discuss some easy set theory, a brief introduction to groups and fields, then abstract functions. After that it becomes more traditional precalc and calc, treating algebraic and trigonometric functions, then introduces limits, some calculus and analytic geometry and a little stat and probability.

Of course in the old days, high school geometry was taught from Euclid, or something similar, and did include proofs. I still recommend this source for geometry (and proof).

 

[thegreenlaser]

It's true that engineering (applied) math is a lot closer to high school math than pure, but in my transition to college I found that there's a key difference in the expectations. In high school, you learn how to solve a certain type of problem in class and you're only expected to solve that particular type of problem. In college, you're expected to solve problems you've never seen before, they just use the same underlying principles as the problems you saw in class. You can't always just memorize the method and regurgitate it on the test. To do well, you have to focus on understanding the theory and how to apply it in new situations. In a sense, you're expected to be able to derive formulas, not just use them.

 

[mathwonk]

This reminds me of the student who complained:

" On the test you asked us to maximize the volume of a closed top box, but in class you only showed us how to do the [harder] case of an open topped box!"

In fact I wonder now if I wasn't being punked by someone from the Daily Show.