Advancing Math Skills: Tips for High School Seniors

[lax1113]

Hey guys,
I am a high school senior and i will be taking AP calculus next year in school. I am very fascinated with math, and kind of want to be ahead of the game, not to prove anything to anyone, but mostly just for myself, to see what level math i can understand. So, with that being said, i was wondering if anyone had any tips on where i could look on the internet to get a further understanding of math, and more advanced concepts. I have taken precalc, so i guess the next step would be calculus, but are there any sites available that give sort of a track to follow in learning calculus?

Thanks,

 

[Defennder]

Websites are not a particular eye-friendly learning resource. Most people would recommend a textbook like Stewart's Calculus.

 

[nicksauce]

Alternatively, you could get a book like "How to prove it" by Velleman, to make sure that your foundations in mathematics and logic are well solidified.

 

[kmikias]

To be honest the best way to study is your textbook.But if u want a website this one will be good because this website helps me in ap exam last year in AB calculus so it should help you (I guess). Normaly called Pauls online Math Notes ... .....http://tutorial.math.lamar.edu/
And finally i recomended a book called Be prepared for the ap calculus exam by Mark Howell and Martha Montgomery. GOOD LUCK.

 

[JasonJo]

I think learning how to prove the things you are using in calculus is pretty important as real math is about proving theorems and not doing plug and chug calculations.

Check out Logic, Proof, and Conjecture. It is a good intro to mathematical proofs textbook, and I think a high school senior could definitely jump into it without any trouble.

Also, if you really want to get crazy, I would also try to learn some linear algebra. Linear algebra is so important and so damn useful, it would definitely be a plus if you at least exposed yourself to some linear algebra. I believe our own mathwonk has some good linear algebra notes on his math department page. Check out his site for the PDF's.

 

[ekrim]

As boring as it sounds, really learning calculus 1 well will be a great asset to you. This may include going ahead of the class in chapters, doing extra problems, and learning more theoretical aspects of what the class is doing.

 

[Focus]

Wikipedia and google will serve you well. I suggest reading on real analysis, as this will give you a good understanding of what calculus is. Or you could take the applied road and look at DEs (differential equations) and such, but I suggest you take the pure path

 

[lax1113]

So, I decided to go out and buy Velleman's how to prove it. So far i have done a chapter and really am surpised how different it is than calculus/precalc with numbers. Last year in physics our teacher would sometime give extra credit questions that would ask to prove something, such as the one time was to prove that when two pool balls hit on a billiard table, that if the friction/spin is negligible, Ball'a' must hit ball'b' and form a right angle. Anyway, that was pretty much the only work that i had ever done with proofs, but i am finding them very enjoyable in that they are sort of like logic puzzles.

Thanks for the suggestion Nick!

 

[lax1113]

Alright, so now I'm a bit further into the book and I'm happy that I'm starting to learn this now before i get thrown into it at a college level. I don't know if its something that gets easier as it is done more, or if i am just not as good at proofs as i am at the math that i have done before, but some of these that according to the book are relatively easy seem very strange and difficult to me. Anyone else have difficulty with proofs when first trying them out?

 

[Howers]

In response to PM,

Saying math proofs are like puzzles is quite accurate. Velleman's book is a good start, so it is good you are working through it. Make sure you do a fair amount of exercises too (7 per section seems good), because reading alone is a waste of time. And spread each section over time, ie. do a section every two days. The material you are dealing with is abstract and you will need a second re-reading the day after (or possibly more) to really see the big picture.

What you are dealing with now is symbolic logic, not mathematics. You are manipulating logical statements to get practice with symbolic notation. This is not something you will be doing in math, but the process of proof is very similar - you resolve your questions into the rules and definitions. Also, my "3 months" is quite arbitrary. Proofs do get easier with time, I will stand by that, but there is no set length of time everything just clicks. The more you put in, the faster you see results. It is a skill you will have to practice daily for a few months, and it is a painful process at first. So it is not surprising that many students find they are seemingly "bad" at math when first exposed to it, despite acing high school. This is a common experience among all students exposed to rigorous mathematics for the first time. I know some very very bright students who even gave up on math, while others that persisted and succeeded in it. So you have to be willing to put in a lot of work, more than any other subject imo, to get anywhere with it.

Pretty much all of math uses proofs (or assumptions). Everything you know about math is the result of someone showing the result is true. For instance, did you know the exponent laws can be proved? (that might be a good exercise, then check on the internet for answers). If you take honors math classes in college, they will all be very proof oriented, right from year 1. Non-honors classes will be like high school, just calculations (this is useful to engineers or non-theoretical scientists, who need more time to devote to their main courses). Proofs are especially prevelant in university algebra classes, including Linear algebra, abstract algebra, and number theory. Real analysis and topology, as well as most third year+ math classes will be all proofs.

As for careers that do proofs for a living, I would imagine the vast majority are in academia (ie. teaching). Obviously, the mathematicians will be doing proofs for a living. Physicists don't really do "math proofs", they instead manipulate equations and derive results that correspond to physical phenomena. Theoretical physicists will have good grounding in proofs because of the heavy math courses they took, but it is not something they will be doing in physics. Philosophers and logisticians do proofs as well, although of a different nature. Computer science too has its share of proofs. In fact, most programming languages are written in a very rigorous fashion similar to that of a mathematical proof. This is why a lot of computer scientists make great mathematicians. As you can see, proof is a very general term. All it really means is presenting an argument. English majors and historians do this all the time, as does most of science. But only in math will you find it so precise and accurate.

Since you are in high school, I recommend you take a course on geometry or study Euclidean geometry on your own. A good book is "Geometry, 2nd ed" by Harold Jacobs. Classical geometry is where proofs were born, so it is a good place to begin either concurrently or after Velleman. An introductory to Linear Algebra will also be beneficial, but save that for after you do calculus. Another book I recommend is Courant's "What is Mathematics"?, a book that has some proofs and will surely make you love math.

 

[qspeechc]

I agree mostly with you Howers, but I do not believe you should waste your time with geometry. The reasonig there is not useful anywhere else, and neither are the results. Geometry in high school is all about find the correct chain of identities to get the result, like a maze. Higher level maths is not like that- it's not hit and miss.

 

[PieceOfPi]

When I took linear algebra right after I graduated from high school (i.e. right after AP Calculus), I was surprised to see so many proofs that my professor covered in the class, since I've never had an experience in writing proofs before. I was shocked and scared; I still got an A in the class (since everybody else in the class did poorly), but I certainly have wished that I had some experience with proofs before.

Later, I took a few courses that uses proofs at somewhat basic level (i.e. number theory and elementary real analysis), and now I find mathematical proofs to be enjoyable.

So the thing you're doing right now is very good, since AP Calculus doesn't do any proof at all. Also, it's also a good sign if you find those proofs to be "very enjoyable", because I've seen some math majors who "hate proofs" because they thought studying math was like studying AP calculus.

 

[lax1113]

Howers, can't tell you how much i appreciate your responses. Also, pieceofpi, I am actually not planning on majoring in math, most likely engineering, but i am just very interested in math so i wanted to get to more advanced math than just simply calculations. Majoring in math is still definately an option, as that i really do enjoy it, I just don't know what career lies in a math major, other than teaching, which I am not in the least bit interested in.

 

[PieceOfPi]

Howers, can't tell you how much i appreciate your responses. Also, pieceofpi, I am actually not planning on majoring in math, most likely engineering, but i am just very interested in math so i wanted to get to more advanced math than just simply calculations. Majoring in math is still definately an option, as that i really do enjoy it, I just don't know what career lies in a math major, other than teaching, which I am not in the least bit interested in.

I see, then you'll possibly be way ahead of the other engineering students when it comes to math

BTW, when I started out college, I was a biochemistry major, thinking of becoming a medical researcher, and find a cure for cancer or AIDS so I can give some kind of hypocritical speech at Novel Prize Award Ceremony (e.g. "I've got frustrated so many times, but thinking of the people who are suffering from this disease always helped me work hard!"). But as time past, I found math to be much more interesting than biochemistry, so I switched to math, and thinking of double majoring in something math-related as well (e.g. physics, computer science, econ, etc). I may not be able to give that speech anymore, but I certainly like the choice I made so far.

And I agree, I'm not so sure what I can do with math major other than teaching or work in financial sector (which is NOT something I want to dol). And that's why I'm considering of double major.

 

[lax1113]

Piece of pi, I'm really in a similar boat as you. Theres no doubt that math is what i like, more than any other subject, but the bottom line is, a career should be something you love, but at the same time, it needs to pay the bills. I am hoping that engineering will be enough math oriented to keep me happy, and who knows, maybe after some engineering i can go back to school to learn more math, just you know, for the hell of it.

 

[Focus]

Piece of pi, I'm really in a similar boat as you. Theres no doubt that math is what i like, more than any other subject, but the bottom line is, a career should be something you love, but at the same time, it needs to pay the bills. I am hoping that engineering will be enough math oriented to keep me happy, and who knows, maybe after some engineering i can go back to school to learn more math, just you know, for the hell of it.

I think you will be quite disapointed with the level of maths in engineering. Here in UK most maths graduates go on to investment banks in risk management or research. It does pay very well I have to say. You always could do a maths undergrad and do engineering masters. If you truly are interested in maths then you should do it.

 

[Defennder]

I think you will be quite disapointed with the level of maths in engineering. Here in UK most maths graduates go on to investment banks in risk management or research. It does pay very well I have to say. You always could do a maths undergrad and do engineering masters. If you truly are interested in maths then you should do it.

I doubt it's possible. You need to know a lot of undergrad engineering before you'll be able to do grad engineering courses. A math major, even an applied math major would simply not be enough. If you crave math, the best way would be to either double-major in it, or minor in it.