Abstract Numbers in their
natural dependence on sets of visible things are abstract products
which result from ignoring nature of elements of these sets and
any kind of their organization (the way how they are arranged,
grouped, etc). We call formulation of this way of cognition {ıt Cantor
principle of invariance of number\/} and, in this paper, we apply it
to a coherent exposition of the first grade arithmetic.
Reacting to the way how the elements of a set are grouped, one
expression is written, regrouping these elements, another
expression is written. Then, such two expressions are equated
since they denote one and the same number. Thus, this interplay
between meaning and symbolic expressing is the ground upon which
the range of numbers up to 20 is structured.
Two disjoint sets together with their union make an additive
scheme to which an addition or a subtraction task may be attached.
Dependently on such a task, sums and differences are written as
expressions denoting numbers. To reach the unique decimal (in
digits) denotation of a number some steps of transforming the
corresponding expressions are made. Displaying these intermediate
steps leads to a thorough understanding of the arithmetic
procedures, which should precede their suppressing which leads to
automatic performance. In particular, methods of adding and
subtracting when the tenline is crossed are treated in detail.
