3
$\begingroup$

I am a junior in high school currently in multivariable calculus (online) and find it kind of boring since it's similar to Calc BC from last year, emphasizing applications of math to, for example, engineering rather than theory. I am highly interested in the theory of pure mathematics and love the rigor and proofs of higher math, and am constantly watching lectures and trying to learn as much as I can of these subjects. There are not really any teachers at my school that can help me with math beyond single variable calculus. However, I took the AIME math competition last year and got a 2, showing I might not have as strong of a base of mathematics as I need to prepare me for higher math later in college/grad school.

I've currently been working through a number theory book and Linear Algebra Done Right by Sheldon Axler, but I also have a less abstract (but still challenging) Art of Problem Solving: Geometry book. Should I devote more time at this point in my mathematical career to these fundamentals provided by Art of Problem Solving, or would I be fine to skip those and continue working on the more theoretical textbooks? (I also would really like to work through Calculus by Spivak or Apostol, though I'm not sure which is better.)

Any advice would be greatly appreciated.

$\endgroup$

closed as off-topic by Jean-Claude Arbaut, JonMark Perry, mrf, Rob Arthan, Lord_Farin Oct 9 '15 at 23:00

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "Seeking personal advice. Questions about choosing a course, academic program, career path, etc. are off-topic. Such questions should be directed to those employed by the institution in question, or other qualified individuals who know your specific circumstances." – Jean-Claude Arbaut, mrf, Rob Arthan, Lord_Farin
If this question can be reworded to fit the rules in the help center, please edit the question.

  • 1
    $\begingroup$ For what it's worth... if I could do it all over again, I would learn calculus through Spivak's text and multivariable calculus through Ted Shifrin's text. $\endgroup$ – Clarinetist Oct 9 '15 at 15:50
  • $\begingroup$ had you ever heard about axiomatic deductive construction of mathematical theories? in particular, how the group's theory evolves through 200 years just to today's the state-of-the-art branches? $\endgroup$ – janmarqz Oct 9 '15 at 15:55
  • 1
    $\begingroup$ I certainly advocate mastering the fundamentals before going further. You have plenty of time ahead of you, and it sounds like you learn at a fast than average pace as it is. So you're not in danger of never getting to advanced subjects. But I've encountered lots of students who have "seen" all the prerequisites before but simply didn't understand them deeply, and that presents a huge obstacle to learning anything new. At this stage, number theory/linear algebra/geometry/problem solving sounds like the exact right place to be. $\endgroup$ – Greg Martin Oct 9 '15 at 16:16
  • 1
    $\begingroup$ I should point out that doing well on competitions like the AIME is neither necessary nor sufficient for doing well in pure mathematics. I know many people in pure mathematics who loved and excelled in such pursuits, and just as many who disliked them and never put much into them. $\endgroup$ – JHance Oct 9 '15 at 16:54
  • 1
    $\begingroup$ Let's see what the biggest benefits of the two paths are: Problem solving in general - get a feel what to use when, or to expand the number of tools at your disposal by going into more theory. I could be wrong though. I used my knowledge of some areas of math and some simple engineering classes in more advanced engineering classes. The interesting thing is that I just(internally) disregarded the explanations in the classes and just saw it as a combination of the two earlier fields and it went well. So both aspects, the toolkit and the skill to see connections can come in handy! $\endgroup$ – WalyKu Oct 9 '15 at 17:15
2
$\begingroup$

I totally agree with the comments of Greg Martin regarding the absolute need to really master the fundamentals and understand them deeply. I got through four semesters of undergraduate calculus with solid A's for each semester by knowing how to solve the problems presented. Unfortunately, I didn't truly understand anything. When I got to my engineering dynamics course, where we had to actually apply calculus to solve real world problems, everything fell apart and I was totally lost. Get to the point where you understand the fundamentals deeply enough that you can explain them to your mother. Also, don't ignore the applied mathematics side. This is where a lot of the true understanding comes in.

$\endgroup$
  • 1
    $\begingroup$ By fundamentals, are you referring to problem solving provided by the Art of Problem Solving geometry, algebra, and counting/probability/number theory books? Or a firm foundation of calculus and other more advanced topics by more thorough books? $\endgroup$ – MBP Oct 9 '15 at 16:51
  • $\begingroup$ My question was pertaining more to whether it was necessary to have firm grounding in these problem solving and somewhat competition math-based books, or if I could just move onto theory of calculus and linear algebra without worrying too much. $\endgroup$ – MBP Oct 9 '15 at 16:58
  • $\begingroup$ I wish I could comment on the "Art of Problem Solving" as to whether it would contribute to your mastering of the fundamentals, but I am not familiar with the series. My answer, I suppose, was a generalized encouragement for you to embrace the applied mathematics side of math. Studying and solving real-world problems is an excellent way to master and understand the fundamentals. $\endgroup$ – J Erbes Oct 12 '15 at 17:37

Not the answer you're looking for? Browse other questions tagged or ask your own question.