Important part of the school mathematics of XX century
was about geometry and trigonometry.

In particular, students had to remember the Euclid's axioms.

(Especially doubtful is the axiom about measuring of angles that
in implicitly used in the most of courses, but is not even formulated.)

Such a way had sense in century XIX, before the development
of linear algebra, analytic geometry, functional analysis and before computers.

Now I consider such an approach to the teaching (and learning)
of mathematics as non-efficient and outdated.

I suggest the scheme below, although I understand,
that it will take some time to convince at least one educational organization to accept such a program.

The modern course of the basics of the modern mathematics should include the following topics:

0. Mathematical logic. Principles of Boolean algebra. Mathematical notations.1. Natural numbers. Operation ++, addition and multiplication. Axioms.

2. Integer numbers. Operations --, subtraction, division. Combinatorics.

3. Rational numbers. Principles of Algebra. Functions. Practice.

4. Sequences and series. Fundamental sequences. Real numbers. Limits.

5. Algebraic functions. Graphics. Equations. Functions of two variables.

6. Derivatives. Physical applications. Integrals.

7. Fundamentals of the differential equations. Basic elementary functions.

8. Fundamentals of the integrals. Some special functions

9. Basic numerical methods: interpolation, fitting, differentiation, integration, optimization.

10. Fundamentals of the Analytic Geometry. Deduction of axioms of Euclid.

11. Complex numbers. Rules of arithmetics. Practice and the application.

12. Principles of Mathematical analysis. Practice.

13. Axioms of Theory of Probability. Basics of statistics.

14. Use of Mathematics in physics.

The topics #0 and #1 may go parallel,

the topics #5 and #6 may go parallel,

the topics #8 and #9 may go parallel,

the topics #9 and #10 may partially overlap in time,

the topics #10 and #11 and #12 may go parallel,

dependently on the concentration of course and level of comprehension required, and on the total time assigned for the education of mathematics.

Some other topics, especially computation are impotrant for Mathematics. In particular, some of html, postscript, Latex, Mathematica, Matlab, or Fortran, C++ or their surrogates should be tought before #13; in the similar way, students should learn to read, to write, to make some physics, to drive car, to swim, to run, to jump, to climb, to use instruments, to cook, etcetera. But these should not be considered as parts of Mathematics.

Copyleft 2009 by Dmitrii Kouznetsov.

You may use this text for free but you should indicate the author and the source:
**http://www.ils.uec.ac.jp/~dima/2009teachma.html**

------------------------------------------------------