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Advice concerning high school math

  1. Apr 1, 2010 #1
    Hi,
    I am a sophomore in high school who is very interested in math. I have always gotten good grades in math; I am currently in honors precalculus with a 99.6 percent, however I find this class to be exceptionally easy as I am already familiar with the subject matter from algebra 2. My concern is that I am not doing enough to prepare myself for college or have not done enough to be admitted by a good college. Unfortunately, I have done nothing outside of taking high school math courses. I am not a part of my schools math team or any extracurricular activities at the moment, although I do plan to join the math team and the national honors society next year. I don't know for sure what exact field of study I want to enter, but I am interested in majoring in math. My question is what advice or recommendations do you have for a student such as myself.

    I have also read on this website about a general disapproval of high school level math courses and about books that may broaden my understanding of courses I have already taken like geometry and algebra. If anyone could recommend any such books that I may find useful or necessary to prepare for college, I would greatly appreciate that as well.

    Thanks Alot :)
     
  2. jcsd
  3. Apr 1, 2010 #2
    https://www.amazon.com/Basic-Mathematics-Serge-Lang/dp/0387967877
    https://www.amazon.com/How-Solve-Ma...=sr_1_1?ie=UTF8&s=books&qid=1270105995&sr=1-1
    http://www.artofproblemsolving.com/ ---A (possibly) useful forum if you really want to get into competitions or chat with some people with a pretty strong mathematical foundation

    There's a start. A Caveat: Don't let the 'basic mathematics' title make you think it's necessarily simple and easy. If you get through that material you could probably jump to a rigorous treatment of the calculus like those from Spivak or Apostle relatively comfortably. If you can get through all of that before college you'll be way ahead of the game.
     
    Last edited by a moderator: Apr 24, 2017
  4. Apr 1, 2010 #3
    I really like the book Problem-Solving through Problems by Loren Larson. It does a good job of covering all the techniques of proof while also supplying a wealth of interesting and beautiful examples taken from mathematical competitions all around the world.

    A similar book is Putnam and Beyond. This is one fantastic book, but it's also very difficult--it's possibly the hardest book of this kind that I've come across. In short, it lives up to its title. I still recommend checking it out. Even if you find most of it hard to follow right now, it will be a great resource to gradually come back to over the coming years. (And who knows, you might find that you can follow it quite well.)
     
  5. Apr 1, 2010 #4
    High school is really the place to start learning to do proofs, and you sound like you're ready.
    One book that mathwonk recommended is Principles of Mathematics by Allendoerfer and Oakley. I think it's a good place to start. Otherwise there's the book by Velleman: How To Prove It, also good.
    After that, you can read Courant's What Is Mathematics?; or Spivak's Calculus; or even Axler's Linear Algebra Done Right.

    Check out https://www.physicsforums.com/showthread.php?t=122924" as well.
     
    Last edited by a moderator: Apr 25, 2017
  6. Apr 1, 2010 #5
    Thank you very much. I will look into all of these books.
     
  7. Apr 1, 2010 #6
    I would like to second the Basic Mathematics book recommendation. Fundamental, yet very challenging and its focus is more on getting you in the right mindset than teaching you a list of facts. I believe this will prepare you more for college-level mathematics than any accelerated advanced honors calculus class your high school might offer. That is not to say that calculus isn't useful, but it's often overrated as a math course (especially by high schools and teachers there), and usually the treatment isn't all that satisfying mathematically (from a purely mathematical standpoint I actually think that calculus should be deferred to second or third year in college once the student has the appropriate foundations, but from an applied/motivational standpoint I see the futility of such an attempt.). Anyway I'll stop my rant about calculus and simply point you towards the article "http://www.artofproblemsolving.com/Resources/AoPS_R_A_Calculus.php"" over at AoPS (it has some good advice for gifted students so even if you do not feel the need to be "warned" you should take a look at it anyway).

    The book Basic Mathematics is written by Serge Lang and some find his writing style unhelpful, but if you major in mathematics and especially if you take graduate math courses, then you're bound to read at least a couple of his books and you better learn to appreciate it.

    Apart from this book you have plenty of other options. You can start studying subjects outside the normal curriculum that you'll need later. For instance if you're interested in seriously learning calculus I would recommend Spivak's https://www.amazon.com/Calculus-Michael-Spivak/dp/0914098918/1" (unfortunately somewhat expensive). Some like Apostel's calculus books and while I see why it seems attractive I have never seen a person who benefited from Apostel's approach. All kinds of neat ideas, concepts and ways to think are provided in the book, but in my experience it doesn't do a good job of conveying them to someone not already a little mathematically mature. Spivak on the other hand does this excellently in my opinion.

    Alternatively Linear Algebra could be interesting. There's Axler's book which many seem to prefer. Personally I'm partial to Halmos' https://www.amazon.com/Finite-Dimensional-Vector-Spaces-P-Halmos/dp/0387900934" which I believe does an excellent job of conveying the central ideas of linear algebra and mathematics in general. It also includes lots of interesting ideas in the exercises (which can be very challenging). Halmos is a master at mathematical exposition and I consider this book to be very good, but I think the reason it may not really have caught on is that it's a little too sophisticated for the average linear algebra student. If you're looking for a good challenge this book will provide it. Another book of interest by Halmos is Naive Set Theory which gives a good exposition of the basics of set theory from a pretty formal point of view. You may need a little more experience with real mathematics before being able to benefit from this, but at some point you'll probably need to know the stuff in this book and then no other book comes near it in terms of usefulness.


    BTW please note that this advice only concerns how to become better at mathematics and prepare for college. You mention whether good colleges will admit you, and many of these things may not seem impressive since it's hard to boast to colleges that you spend you evenings reading books at home. Whether this is an issue is of course up to you. Personally I feel the undergraduate admission competition is extremely silly. It's really not as important as many people make it out to be, and most colleges provide pretty much the same resources and course offerings to undergraduates (at the graduate level it may begin to matter, but still the focus on rankings, ivy leagues and that kind of stuff is still pretty stupid). I got to a certain extent sucked into this trap as a senior and did things that I didn't really enjoy because I felt they would show the real me. I now wholeheartedly regret participating in the stupid competition, but it's hard not to when fellow students, family, teachers, counselors, etc. make it out to be the most important thing in your life.
     
    Last edited by a moderator: May 4, 2017
  8. Apr 1, 2010 #7

    thrill3rnit3

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    Gold Member

    I have read and agree with the "Calculus Trap" article, because I have experienced it firsthand. I rushed through MVC and LA by summer before junior year, and have learned it pretty well. However, as I am required by my school to take a math class, I feel bored and uninspired because I am sitting through classes that seem like Pre-K to me (I am not allowed to skip the lectures). I feel like the challenge isn't there anymore, which sometimes makes me lose interest in the subject matter.

    So, instead of trying to rush through calculus, I suggest you turn into some other branches of mathematics which are VERY interesting. Start with the mathematics of proof and set theory. Once you get the basics down you can start branching out to combinatorics, probability, or number theory. Then try Linear Algebra which involves more abstract thinking and is usually the bridge to higher mathematics.

    Problem solving questions from the USAMO or IMO are also another fun thing to do, but sometimes they can get pretty annoying, especially when the solution is not showing itself to you, and you have to find it yourself.
     
  9. Apr 1, 2010 #8
    Two other books I would recommend are "Fearless Symmetry: Exposing the Hidden Patterns of Numbers" and Zeitz's "The Art and Craft of Problem Solving" Something that I think makes math more fun is to get "off the beaten path." There is a standard path for teaching math, while it's not a bad thing, you don't see some of the fun stuff that's off the path.

    Also "learning math" and "getting into college" are two different things. Both undergraduate admissions and graduate admissions are something of a weird game. You'll have to play that silly game, but part of playing the game involves figuring out how not to let it get in the way of learning things.
     
  10. Apr 1, 2010 #9
    Threads like this make me think I'm still seriously unprepared for calculus despite my recent attempts at remedying my math gaps.
     
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