Advice for a high schooler in mathematics

[shezi1995]

I am doing A levels and in my first year (equivalent to a junior in american system). I am very passionate about mathematics and physics but I get bored in my a level studies especially maths though I am a topper of my class. I therefore watch online university lectures like those of mit and other open courseware sites to quench my thirst of maths. My plan is to study mathematics at university and then research(I know how different maths is at university) So I want advice on how to study maths keeping in my mind my future goal.

• 
  What are the high school maths topics that I should master and how do I do that? The normal school books are not interesting and are rather easy.

• 
  What should I study? I am also participating in my country's national level math olympiad which leads to participation in the IMO. So should I work hard studying olympiad level maths like number theory and geometry? How much will it benefit me in mathematical career?

• 
  I am finding combinatorics hard. I am more comfortable in algebra and geometry. Is that okay? How do I improve my understanding of combinatorics?

• I also have a copy of calculus by apostol and I like it a lot as I have some experience in proofs due to the olympiad. I have studied few topics and absolutely loved those. Should I continue to study it? Some people here suggest that it is better to approach calculus after one has mastery of algebra and other topics.

 

[espen180]

I would say that coming out of high school you should at least have no problems manipulating algebraic expressions, and should be familiar with trigonometric, exponential and logarithmic functions.

Number theory and geometry will definitely benefit you (especially the latter!) I suggest you learn as much geometry as you can. (Euclid's Elements are freely available online).

Improving your combinatorics would also help, mostly for number theory and abstract algebra. Speaking of which, you can probably pick up a copy of "Visual group theory" immediately and start studying from it to get a head start in abstract algebra.

 

[mathwonk]

maybe try euclid elements, euler's elements of algebra, and apostol's calculus.

 

[phospho]

I am doing A levels and in my first year (equivalent to a junior in american system). I am very passionate about mathematics and physics but I get bored in my a level studies especially maths though I am a topper of my class. I therefore watch online university lectures like those of mit and other open courseware sites to quench my thirst of maths. My plan is to study mathematics at university and then research(I know how different maths is at university) So I want advice on how to study maths keeping in my mind my future goal.

• 
  What are the high school maths topics that I should master and how do I do that? The normal school books are not interesting and are rather easy.

• 
  What should I study? I am also participating in my country's national level math olympiad which leads to participation in the IMO. So should I work hard studying olympiad level maths like number theory and geometry? How much will it benefit me in mathematical career?

• 
  I am finding combinatorics hard. I am more comfortable in algebra and geometry. Is that okay? How do I improve my understanding of combinatorics?

• I also have a copy of calculus by apostol and I like it a lot as I have some experience in proofs due to the olympiad. I have studied few topics and absolutely loved those. Should I continue to study it? Some people here suggest that it is better to approach calculus after one has mastery of algebra and other topics.

If you go through C3 and C4 now, and enter for STEP I this year then you'll be in good stead for at least a cambridge interview. Studying for STEP itself improves your mathematical maturity.

 

[shezi1995]

Thank you everyone. I would certainly try Euclid's elements. And perhaps would check out the STEP I questions as well since its also more directly related to A level.
I do have my question regarding the olympiad maths. Should I work hard in that? Will it be of great help?

 

[phospho]

Thank you everyone. I would certainly try Euclid's elements. And perhaps would check out the STEP I questions as well since its also more directly related to A level.
I do have my question regarding the olympiad maths. Should I work hard in that? Will it be of great help?

I would, but remember that higher UMS during your AS levels is more beneficial than a competition. But definitely compete and see how far you can get.