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Now I consider such an approach to the teaching (and learning) of mathematics as non-efficient and outdated.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