Mathematics as a science, viewed as a whole, is a
collection of branches.

The largest branch is that which builds on the
ordinary whole numbers, fractions, and irrational numbers, or what,
collectively, is called the real number system.

Arithmetic, algebra, the study of functions, the
calculus, differential equations, and various other subjects which follow the
calculus in logical order are all developments of the real number system.

This part of mathematics is termed the mathematics of
number.

A second branch is geometry consisting of several
geometries.

Mathematics contains many more divisions.

Each branch has the same logical structure: it begins
with certain concepts, such as the whole numbers or integers in the mathematics
of number, and such as point, line and triangle in geometry.

These concepts must verify explicitly stated axioms.

Some of the axioms of the mathematics of number are
the associative, commutative, and distributive properties and the axioms about
equalities.

Some of the axioms of geometry are that two points
determine a line, all right angles are equal, etc.

From the concepts and axioms theorems are deduced.

Hence, from the standpoint of structure, the concepts,
axioms and theorems are the essential components of any compartment of
mathematics.

We must break down mathematics into separately taught
subjects, but this compartmentalization taken as a necessity, must be
compensated for as much as possible.

Students must see the interrelationships of the
various areas and the importance of mathematics for other domains.

Knowledge is not additive but an organic whole and
mathematics is an inseparable part of that whole.

The full significance of mathematics can be seen and
taught only in terms of its intimate relationships to other fields of
knowledge.

If mathematics is isolated from other provinces, it
loses importance.

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