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I am a 13 year old boy from Hong Kong, My dad is a math teacher which probably accounts for SOME of my interest in math. I am really interested in math. I self-taught myself (with little help form my dad when I needed help) Trigonometry, Differential and Integral Calculus (well most of it), Solid of a revolution, Taylor and Maclaurin series, and recently I have been learning some set theory.

I have a feeling that the things I learn are going all over the place, I just learn what i find interesting, and sometimes I don't understand some topics because I don't know some previous topics so I will need to go back to learn them. Should there be some order with what to learn first and what is the order?

Thanks in advance

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  • $\begingroup$ You could try working through the curriculum / books at artofproblemsolving.com. $\endgroup$ – littleO Sep 26 '13 at 9:54
  • $\begingroup$ whats artofproblemsolving.com?? $\endgroup$ – YYC Sep 26 '13 at 10:39
  • $\begingroup$ Www.artofproblemsolving.com is the URL of a website. $\endgroup$ – The Chaz 2.0 Sep 26 '13 at 22:19
  • $\begingroup$ It's actually very helpful to talk to professors and ask for guidance. There are now many enrichment programs. You should ask your math teacher for more info. $\endgroup$ – user27126 Oct 5 '13 at 6:39
  • $\begingroup$ Have you considered selecting an answer to this question? $\endgroup$ – CommutativeAlgebraStudent Nov 2 '13 at 23:28
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I recommend finding some old math tournament problems, there's a million to chose from. Find some at about your level and learn whatever you have to do in order to be able to solve the problems.

If there is a question that interests you that you couldn't solve, figure out what you need to study in order to be able to solve it.

There's a world of difference between "knowing" an area of math, and being able to use it like a weapon.

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I think, at your age, you can play more with Combinatorics, planar geometry and elementary number theory. These fields are full of interesting and exciting results and most of the time, these results have interesting history which gives information about history of mathematics.

It is always good if you study mathematics in a formal way starting from formalism of set theory. But math is not only about the precision and formalism, it is also about having vision and intuition.

I believe that starting with basic combinatorics, geometry, number theory and analysis is a very good way to develop a preliminary vision about what is mathematics and what it is going to do.

Also, solve as many problems as you can!

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artofproblemsolving.com is a website where students learn math with an emphasis on creative problem solving, and also train for math contests. They have developed a curriculum, including books on number theory and counting and probability. The curriculum emphasizes combinatorics, planar geometry, and elementary number theory, as Arash suggested.

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You should start with linear algebra (vectors, matrices, linear transformations, vector spaces), then move on to abstract algebra, and from there there's a lot of places you could jump to. A great abstract algebra book with heavily linear algebraic approach is Michael Artin's Algebra.

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  • $\begingroup$ I'd suggest set theory, groups, rings, modules, matroids, then fields and abstract vector spaces. With topology-in-between. Taylor theorem to understand linear independence and dimensionality. With polynomials and functionals, linear algebra is so much more sense! $\endgroup$ – Brian Cannard Jun 24 at 16:35
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Since you come from China, I think these good series of lecture notes are suitable for you---基础数学讲义 (基礎數學講義, Lectures in Fundamental Mathematics) by 项武义 (項武義, Wu-Yi Hsiang). The version I have is written in simplified Chinese (I don't know whether it has traditional Chinese version). It has five volumes. They are

  1. 基础代数学 (基礎代數學, Basic Algebra) concerning polynomial and little linear algebra;

  2. 基础几何学之一 (基礎幾何學之一, Basic Geometry I) concerning on classical geometry;

  3. 基础几何学之二 (基礎幾何學之二, Basic Geometry II) concerning on analytic geometry;

  4. 基础分析学之一 (基礎分析學之一, Basic Analysis I) concerning analysis in one variable;

  5. 基础分析学之一 (基礎分析學之一, Basic Analysis I) concerning analysis in several variable.

I think these are suitable for you. You may find some sources by searching.

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  • $\begingroup$ What's more, I think you should learn some more algebras then what these lectures present. $\endgroup$ – Cubic Bear Apr 28 '18 at 10:43

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