(Soft) What maths should I concentrate on at 16-18 years old? [closed]
Some background information first of all: I'm 16 now and just started studying mathematics intensely. I live in the UK and my goal is to eventually become very good at advanced mathematics (graduate level and up) but that is still far in the future. I'm about a year or slightly more ahead from the expected curriculum knowledge in my age group.
The advice I'm seeking of is that I am not sure whether to keep going ahead of the curriculum and eventually learning university maths a year before actually going to uni or should I try to improve my problem solving skills by for example trying out Mathematical Olympiad questions. If I do MO instead then I fear I might be wasting time (even if I'm improving at problem solving) when all I really want to do is the more advanced stuff which is more proof laden I hear anyway.
To answer the question it might be helpful to know what my intentions for pursuing mathematics are. I enjoy learning new fields of mathematics that I did not know of before. A new topic excites me, a new problem to solve with the same tools just doesn't have the same effect on me mentally. I can pick up stuff quiet quickly and get bored if I'm forced to keep doing what I already know without any new knowledge. I realise however that mathematical ability is not just about knowledge but also problem solving skill.
Basically, if you're just interested in what topics to learn next, you really could start on almost anything. Your choices roughly come down to
- foundational math (set theory, logic, proofs)
- geometry
- linear algebra
- calculus/ real analysis (at the level of Spivak)
- abstract algebra
- number theory
- probability theory
Each has their benefits. I'd suggest taking some time to research each one and trying to figure out which grabs your attention the most. Then you can invest a more substantial amount of time and figure out if it really is someone you enjoy. If not, you've definitely got time to change your mind. The point though is to just find something you like and run with it.
It would be nice if you could learn some basics of calculus.
But be sure first that you are already good at the very basics of algebra, identities, logarithms etc.
Do not try Math Olympiad problems.
In my opinion, they train you to be a good "warrior" for 3 hours (the exam) but a mathematician is not a warrior and certainly has plenty of time.
Try to understand the proofs you read and not only memorize the theorems.
Along with Calculus a good advice would be to study Number Theory.
This way you will see what mathematical genius actually is. Trust me!
I would recommend reading Velleman's How to Prove it, which teaches you about how to think logically and construct mathematical proofs. This is often the biggest hurdle for new mathematics students: figuring out what constitutes a proof, what are logically valid arguments, and how to present a proof.
Alongside Velleman, I would recommend Axler's Linear Algebra Done Right. Linear algebra is the study of vector spaces (and later on, modules) and the linear transformations between them. Linear algebra plays a key role in every field of mathematics, as well as in physics and computer science. For example, the derivative operation in calculus is all about linearizing functions.
Nothing is wasted. Hard problems which force you to think are always good. Cambridge University has a reading list and there is more information on the website. And I guess other faculties will do something similar.
My personal experience in the US is that there is a vast difference between the typical high school curriculum (algebra, calculus which are primarily learning methods) and what might be considered pure math.
The ability to prove things (as several others have mentioned) is paramount. This also entails recognizing what conditions are necessary and sufficient, where thins can fall apart, and a wonderful exercise in self-reflection to consider if you really know what you are doing.
That said, I feel (once you have a familiarity with calculus, also mentioned above) that you persue a particular course. Many programs begin with real analysis. The benefit of this is that math knowlegde is often cumulative, and thus following a course will have the added benefit of seeing how and where prior developed material comes into play.
This is quite different from a non-math or science program, as you can read any Shakespearian play or Wordsworth poem without any prior experience.
If you are interested, here is a link to a free down-load of a RA course given by Vaughan Jones (Fields Medal). They are self-contained meaning they start at the beginning, and you need not look elsewhere as you progress. (Notice however the first few pages are a bit technical for thoroughness, so take a glance, and start a page 4.) There is nothing like the real thing.
https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxtYXRoMTA0c3AyMDExfGd4OjJiNTJkM2M2ZWUzZGIwYWQ